The impact of thin idealized media images on body satisfaction
Body Image 10(2013)509–514
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Body
Image
j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /b o d y i m a g
e
The impact of thin idealized media images on body satisfaction:Does body appreciation protect women from negative effects?
Emma Halliwell ?
Centre for Appearance Research,Department of Psychology,University of the West of England,Frenchay,Coldharbour Lane,Bristol,BS169QY,UK
a r t i c l e i n f o Article history:
Received 13December 2012
Received in revised form 16July 2013Accepted 17July 2013Keywords:
Positive body image Body appreciation Media
Body dissatisfaction Women
Experimental
a b s t r a c t
This article examines whether positive body image can protect women from negative media exposure effects.University women (N =112)were randomly allocated to view advertisements featuring ultra-thin models or control images.Women who reported high levels of body appreciation did not report negative media exposure effects.Furthermore,the protective role of body appreciation was also evident among women known to be vulnerable to media exposure.Women high on thin-ideal internalization and low on body appreciation reported appearance-discrepancies that were more salient and larger when they viewed models compared to the control group.However,women high on thin-ideal internalization and also high on body appreciation rated appearance-discrepancies as less important and no difference in size than the control group.The results support the notion that positive body image protects women from negative environmental appearance messages and suggests that promoting positive body image may be an effective intervention strategy.
?2013Elsevier Ltd.All rights reserved.
Introduction
Recently,body image researchers have paid increasing atten-tion to the experience of positive body image.This attention has been prompted by a desire to expand our understanding of body image beyond the prevention of distress and to foster the promo-tion of body appreciation (Tylka,2011).As a consequence,there has been a ?urry of research activity around positive body image.Positive body image is conceptualized as an overarching way of experiencing the body that encompasses love and respect (Wood-Barcalow,Tylka &Augustus-Horvath,2010),and this construct has been operationalized in research as body appreciation (Avalos,Tylka,&Wood-Barcalow,2005).
The Body Appreciation Scale (Avalos et al.,2005)taps into four essential qualities of positive body image:holding positive eval-uations of the body,body acceptance,respecting and attending to bodily needs,and protecting the body by rejecting unrealistic appearance ideals.Body appreciation is not simply the experience of body satisfaction,rather it is a way of valuing one’s body and orientating cognitive processing to protect and promote a positive view of the body.The conceptual distinction between body appre-ciation and body dissatisfaction has been supported empirically (Avalos et al.,2005).Positive body image incorporates a protec-tive processing style;therefore,it is expected that women with
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positive body image avoid in?uences that have a potentially detri-mental impact.Consistent with this assertion,there is correlational evidence that body appreciation is associated with low levels of consumption of appearance focused media (Swami,Hadji-Michael,&Furnham,2008)and is negatively correlated with internalization of sociocultural ideals (Tylka &Kroon Van Diest,2013).Moreover,qualitative evidence suggests that adolescent girls with positive body image are highly critical of unrealistic appearance ideals in the media (Holmqvist &Frisén,2012).However,to date,no experi-mental research has examined the claim that women with positive body image are protected from negative environmental messages around appearance.
One of the most potent sources of sociocultural messages about appearance is the media.Meta-analyses con?rm that exposure to thin bodies idealized in the media has a small to moderate neg-ative effect on body-and weight-dissatisfaction,negative affect,internalization of the thin-ideal,and eating behavior among ado-lescent girls and young adult women (Grabe,Ward,&Hyde,2008;Groesz,Levine,&Murnen,2002;Want,2009).In addition,indi-vidual studies suggest that this negative exposure effect is also experienced by women in later adulthood,and by girls as young as ?ve (Dittmar,Halliwell,&Ive,2006;Dittmar &Howard,2004;Halliwell &Dittmar,2004).A number of individual difference fac-tors have been shown to moderate these media exposure effects.Two meta-analyses have found that women who report body sat-isfaction and who have not internalized the thin beauty ideal are protected from media exposure effects (Groesz et al.,2002;Want,2009).Although body appreciation is related to body satisfaction
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510 E.Halliwell/Body Image10(2013)509–514
and thin-ideal internalization,it is a conceptually and empirically distinct construct(Tylka,2011).Therefore,it represents an addi-tional potential protective factor against negative media exposure effects.
As well as moderating media exposure effects,thin-ideal inter-nalization is a key risk factor for the development of negative body image and disordered eating(Stice,Ng,&Shaw2010;Thompson& Stice,2001).Although all women will be aware that cultural ideals of attractiveness value thinness for women,not all women buy into this dominant ideal.Thin-ideal internalization refers to the extent that someone incorporates sociocultural ideals for women’s beauty into their own personal ideals and values(Thompson,Heinberg, Altabe,&Tantleff-Dunn,1999).In prospective studies,thin-ideal internalization predicts negative body image and disordered eat-ing,yet not all women who internalize the thin-ideal develop eating disorders(Stice et al.,2010).Women who buy into sociocultural ideals of attractiveness but who also accept their bodies despite not matching this ideal and view their bodies with respect and love maybe less likely to experience the negative consequences of thin-ideal internalization.Research has not yet examined the inter-action between body appreciation and known risk factors for body dissatisfaction and eating disorders.
The current study examines whether body appreciation protects women from negative media exposure effects.In addition,it exa-mines the interaction between positive body image and thin-ideal internalization.Women who are high on thin-ideal internalization are known to be vulnerable to negative exposure effects.One key feature of body appreciation is processing sociocultural messages about appearance in a self-protective way.Therefore,high levels of body appreciation may change the way women process images of idealized female models and may protect women who internalize the thin-ideal from experiencing negative effects of viewing media images.
Consistent with previous research?ndings,it is hypothesized that thin-ideal internalization will increase women’s vulnerability to media exposure.Speci?cally,women who are high on thin-ideal internalization will report more negative state body image after viewing models than control images.For women low on thin-ideal internalization,there will be no difference between conditions on outcome variables.
It is hypothesized that body appreciation will protect women from media exposure effects.For women with high levels of body appreciation,there will be no difference in measures of state body image after viewing images of idealized models or control images. In contrast,women low on body appreciation will report more negative state body image after viewing model images than after control images.
Furthermore,it is hypothesized that the protective nature of body appreciation will also be evident among women who inter-nalize the thin-ideal.For women high on thin-ideal internalization and also high on body appreciation,there will be no difference between state body image after viewing models or control images. Yet,women high on thin-ideal internalization but low on body appreciation will report more negative state body image after view-ing models than control images.
Method
Participants
Female psychology students(N=130)enrolled in a large uni-versity in the United Kingdom were recruited to take part in a study on“Attitudes to Advertising”in return for course credits. Complete data were obtained for112women;49women in the control condition and63in the model condition.The majority of participants(87%)were White,their mean age was20.04years (SD=2.61,range=18–39),and their mean BMI was22.33(SD=2.99, range=14.90–34.80).
Advertising Images
There were?ve print advertisements in each condition.The adverts in the model condition were selected from popular women’s magazines and showed full body shots of thin mod-els advertising perfume,clothes,and a soft drink.In the control condition,product-only advertisements for makeup,perfume,and handbags were selected from popular women’s magazines. Measures
Positive body image.The Body Appreciation Scale(Avalos et al.,2005)was used to assess positive body image.Participants responded to13items on a5-point scale ranging from never(1)to always(5).The scale has a unidimensional structure and demon-strated evidence of reliability(internal consistency and3-week test–retest)and construct validity for women(Avalos et al.,2005). Cronbach’s alpha for the current sample was.92.
Thin-ideal internalization.The internalization subscale of the Sociocultural Attitudes Towards Appearance Questionnaire (SATAQ-3:Thompson,van den Berg,Roehrig,Guarda,&Heinberg, 2004)was administered.Respondents indicated their agreement with nine statements on a5-point Likert-type scale,ranging from de?nitely disagree(1)to de?nitely agree(5).Construct validity and internal consistency reliability of this measure have been demon-strated for women(Thompson et al.,2004).Cronbach’s alpha for the current sample was.90.
State body image.State body image was measured unobtru-sively through ratings of appearance self-discrepancies.The state version of the Self-Discrepancy Index(state-SDI;Dittmar,Beattie, &Friese,1996;Halliwell&Dittmar,2006)was used.The measure asks participants to complete?ve sentences of the format,“I.... but I would like...,”describing aspects of themselves that they would ideally like to change right now.They then rate each self-discrepancy statement in terms of magnitude,ranging from a little different(1)to extremely different(6)and salience,ranging from a little important(1)to extremely important(6).
Self-discrepancies are then coded into a category of interest(e.g., appearance-related self-discrepancies).The extent and salience of self-discrepancies were measured separately because they can be independent of each other.For instance,there may be a large discrepancy between a person wanting to be“thin”rather than “overweight”,but it may not be important to them.Therefore,for each appearance-discrepancy two scores were calculated.The size of the state appearance-discrepancies is calculated from the sum of all the size ratings assigned to discrepancies coded as appearance-related.The salience of appearance-discrepancies is the sum of the importance rating assigned to discrepancies coded as appearance-related.The measure is additive because respondents had the opportunity to write about anything concerning themselves,list-ing more appearance-related aspects is an indication of larger and more salient discrepancies.The possible range of both the size and salience rating are0–30.Zero means the person did not list any self-discrepancies relating to appearance.Thirty means that all?ve of their listed self-discrepancies were related to appearance and had magnitude or importance ratings of6.
The self-statements were coded as appearance-related discrep-ancies if they explicitly referred to appearance(e.g.,my weight,my boobs),as opposed to other aspects of the self(e.g.,my?nances,
E.Halliwell/Body Image10(2013)509–514511
Table1
Pre-exposure BMI,age,thin-ideal internalization,and body appreciation by condition.
Measure Model,M(SD)Control,M(SD)t-value df p-value
BMI22.03(2.92)22.40(3.48)0.60107.55
Age19.94(1.62)20.18(3.46)0.78106.44
Thin-ideal internalization 3.36(0.73) 3.13(0.72)?1.76113.08
Body appreciation 3.17(0.67) 3.24(0.70)0.64106.64
Note:N=112;n model condition=63,n control condition=49.
my love life).Two coders independently coded the data,and there was100%agreement between the coders.
The SDI has good convergent validity with other measures of women’s and men’s body image and criterion-related validity with eating behaviors(Halliwell&Dittmar,2006).The state-SDI is also appropriate for assessing media exposure effects among under-graduate women and men(e.g.,Dittmar&Halliwell,2007;Dittmar, Halliwell,&Stirling,2009).
Advertising evaluation.To support the cover story,women were asked to rate their impression of the advert,their impression of the brand,and how much they liked the advert.
Procedure
The study was approved by the University’s Ethics Committee.It was advertised as a study investigating women’s attitudes towards advertising and personality factors in?uencing adverting effective-ness.On agreeing to take part,women were randomly allocated to a condition via a computer generated randomization table.Partic-ipants were then sent a link to an online questionnaire in which they were asked to complete at their convenience.They were told that personality,mood,and thoughts about oneself in?uence pre-ferences for advertising and that the online questionnaire included questions relating to these factors.They completed measures of body appreciation and thin-ideal internalization as well as?ller items about personality and consumer attitudes to disguise the true focus of the study.Women also reported their age,height, weight and ethnicity.One week later participants attended a lab session.Initially,65women were randomly assigned to each condi-tion.However,18of these women did not attend the experimental session due to illness,transport problems,and an unplanned uni-versity closure which reduced the sample size to112.Women were tested individually.They rated the advertisements at their own pace and then completed the appearance-discrepancy measure.At the end of the experiment,women were asked to state the pur-pose of the study in their own words in order to check that they had believed the cover story.All of the women were na?ve to the actual aims of the research.The participants were thanked and, when data collection was complete,they were emailed a written debrief describing the true aims of the study.They were also given the opportunity to retract their data from the study.None of the participants asked for their responses to be removed.
A post-test only design was chosen to avoid setting up demand characteristics.Some researchers have successfully used Visual Analogue Scales(VAS)in pre-and post-test designs(e.g.,Birkeland et al.,2005),and VAS scales are well suited to repeated measure-ment designs.However,when using more detailed measures of body image,repeated measurement is likely to make the aims of the study apparent e.g.,participants may be able to remember,and reproduce,their pre-test responses or shape their answers accord-ing to what they think the experimenter wants to?nd(Thompson, 2004).Thus,when participants suspect the true purpose of media exposure studies,the identi?cation of adverse effects is more likely (Mills,Polivy,Herman,&Tiggemann,2002).Mills et al.(2002)sug-gested that,ideally,body image measures should be presented as part of a separate study to eliminate demand characteristics.How-ever,the naivety check in the present research con?rmed that the use of an unobtrusive measure of appearance concerns combined with the post-only design did not set up demand characteristics.
Results
t-tests,reported in Table1,revealed that there was no signi?cant difference in BMI,age,thin-ideal internalization,or body apprecia-tion between women assigned to each condition,assuring that the randomization was effective in creating equal groups.Body appre-ciation and thin-ideal internalization were moderately correlated, r=?.36,p<.001,supporting the notion that they are independent factors.High and low groups on thin-ideal internalization and body appreciation were created using a median split on each variable, the median for thin-ideal internalization was3.5and the median for body appreciation was3.27.
Appearance-discrepancy Size
A2(condition)×2(level of thin-ideal internalization)×2(level of body appreciation)ANOVA was used to examine whether thin-ideal internalization and body appreciation,independently or in combination,moderated the impact of media exposure on appearance-discrepancy size.There was no statistically signi?cant main effect of media exposure on appearance-discrepancy size, F(1,104)=0.98,partialá2=.01,p=.33.Surprisingly,neither level of thin-ideal internalization,F(1,104)=0.81,partialá2=.01,p=.37, nor level of body appreciation,F(1,104)=1.43,partialá2=.01, p=.23,had a signi?cant main effect on appearance-discrepancy size.
Consistent with previous research,there was a signi?cant interaction between exposure and thin-ideal internalization,F(1, 104)=3.93,partialá2=.04,p=.05.Among women low on thin-ideal internalization,there was no signi?cant difference in the size of appearance-discrepancies reported in the control condition (M=5.97,SD=4.30)and the model condition(M=5.48,SD=3.82), F(1,52)=.49,partialá2=.01,p=.49,d=0.12.However,women high on thin-ideal internalization reported larger appearance-discrepancies if they had viewed models(M=8.06,SD=4.42) compared to those who had viewed control images(M=5.50, SD=3.33),F(1,52)=4.40,partialá2=.08,p=.04,d=?0.65.
Body appreciation did not moderate the effect of media expo-sure,as there was no signi?cant interaction between media exposure and level of body appreciation,F(1,104)=1.37,partial á2=.01,p=.24.The interaction effect between thin-ideal inter-nalization and body appreciation approached signi?cance,F(1, 104)=3.72,partialá2=.04,p=.06.Central to the current research question,the3-way interaction between condition,level of thin-ideal internalization and level of body appreciation was signi?cant, F(1,104)=11.66,partialá2=.10,p=.001.This?nding suggests that the moderating role of body appreciation on media exposure differs for women depending on their level of thin-ideal internalization.
To interrogate the3-way interaction,analysis was run separately for women according to their level of thin-ideal inter-nalization.The means and SD s for appearance-discrepancy size are
512 E.Halliwell /Body Image 10(2013)509–514
Table 2
Mean appearance-discrepancy size and salience scores in each condition by level of thin-ideal internalization and level of body appreciation.
Model,M (SD )
Control,M (SD )
Appearance-discrepancy size High TII High BA 6.29(3.79)7.50(3.34)
Low BA 9.18(4.51) 3.50(1.84)Low TII High BA 5.24(4.12) 4.29(3.47)Low BA 5.90(3.41)8.33(4.35)
Appearance-discrepancy salience High TII High BA 6.29(3.14)9.10(3.18)Low BA 10.05(4.78) 4.20(1.87)Low TII High BA 4.65(4.05) 4.88(4.33)Low BA 6.60(4.43)
7.08(4.03)
Note :TII,thin-ideal internalization;BA,body appreciation.
reported in Table 2.In the low thin-ideal internalization group,there was no interaction between media exposure and level of body appreciation,F (1,52)=2.53,partial á2=.05,p =.12.In the high thin-ideal internalization group,the interaction between media exposure and level of body appreciation was signi?cant,F (1,52)=10.47,partial á2=.17,p =.01.This interaction is displayed in Fig.1.Women who were high on thin-ideal internalization and low on body appreciation who viewed models reported larger appearance-discrepancies compared to women who viewed control images,F (1,22)=14.55,p =.001,partial á2=.33,d =1.65.However,there was no signi?cant media exposure effect among women high on thin-ideal internalization and high on body appre-ciation,F (1,22)=0.66,p =.42,partial á2=.03,d =?0.34.
In sum,these ?ndings indicate that the size of appearance-discrepancies was only impacted by media exposure for some women.Speci?cally,the size of appearance-discrepancies was not impacted by media exposure for women who are low on thin-ideal internalization or for women who are high on thin-ideal internal-ization but are also high on body appreciation.However,viewing models was associated with larger appearance-discrepancies when women were high on thin-ideal internalization but low on body appreciation.
Appearance-discrepancy Salience
The same analysis was run on appearance-discrepancy salience.Again,there was no statistically signi?cant main effect of media exposure on appearance-discrepancy salience,F (1,104)=0.55,partial á2=.01,p =.46.There was a main effect of
thin-ideal
24681012Model
Control
Low body apprecia?on High body apprecia?on
Fig.1.Appearance-discrepancy size scores for women high in thin-ideal internal-ization by condition and level of body
appreciation.
2
468
1012Model
Control
Low body apprecia?on High body apprecia?on
Fig.2.Appearance-discrepancy salience scores for women high in thin-ideal inter-nalization by condition and level of body appreciation.
internalization,F (1,104)=4.21,partial á2=.04,p =.04,but there was no main effect of body appreciation,F (1,104)=0.93,partial á2=.01,p =.34.In this case,there was no signi?cant interaction between exposure and thin-ideal internalization,F (1,104)=1.44,partial á2=.01,p =.23.Instead,there was a signi?cant interac-tion between media exposure and level of body appreciation,F (1,104)=7.23,partial á2=.07,p =.01.For women high on body appreciation,there was no signi?cant difference in the salience of appearance-discrepancies reported by women who had viewed models (M =5.39,SD =3.70)and women who had viewed control images (M =6.44,SD =4.40),F (1,56)=0.99,par-tial á2=.02,p =.32,d =0.26.In contrast,women who were low on body appreciation and viewed models reported more salient appearance-discrepancies (M =8.97,SD =4.88)than women low on body appreciation who viewed control images (M =5.77,SD =3.49),F (1,52)=6.98,partial á2=.12,p =.01,d =?0.75.
The interaction effect between level of thin-ideal internalization and level of body appreciation was not signi?cant,F (1,104)=2.86,partial á2=.03,p =.09.Similar to the ?ndings with appearance-discrepancy size,the 3-way interaction between condition,level of thin-ideal internalization,and level of body appreciation was signi?cant,F (1,104)=8.11,partial á2=.07,p =.001.
Analysis was run separately for women according to their level of thin-ideal internalization.The means and SD s for appearance-discrepancy salience are reported in Table 2.The interaction between media exposure and level of body appreciation was not signi?cant among women low on thin-ideal internalization,F (1,52)=0.01,partial á2=.00,p =.92.However,the interaction was signi?cant among women high on thin-ideal internalization,F (1,52)=16.85,partial á2=.25,p =.001.This interaction is shown in Fig.2.Women high on thin-ideal internalization but low on body appreciation reported more salient appearance-discrepancies in the model condition than in the control condition,F (1,30)=13.80,partial á2=.32,p =.001,d =1.61.Interestingly,women high on thin-ideal internalization and also high on body appreciation reported less salient appearance-discrepancies when they viewed models than when they viewed control images,F (1,22)=4.63,partial á2=.17,p =.04,d =?0.89.
These ?ndings indicate that viewing media models,compared to control images,did not signi?cantly impact on the salience of appearance-discrepancies among women low on thin-ideal inter-nalization.The combination of high thin-ideal internalization and low body appreciation was associated with negative media expo-sure effects because viewing models was associated with more salient appearance-discrepancies than viewing control images.In contrast,when high thin-ideal internalization was coupled with high body appreciation viewing models was associated with less
E.Halliwell/Body Image10(2013)509–514513
salient appearance-discrepancies compared to viewing control images.
Discussion
This study provides preliminary evidence to suggest that body appreciation protects women from negative media exposure effects.Moreover,this protection is evident among women who internalize the thin-ideal and are known to be vulnerable to body image disturbance.
The?ndings replicate previous research demonstrating that women who strongly internalize the thin-ideal report larger appearance-discrepancies after viewing thin models than control images.Interestingly,exposure to media models did not impact the salience of appearance-discrepancies among this group of women. This suggests that women who internalize the thin-ideal feel more dissatis?ed with their appearance after viewing models,but that they do not place increased importance on this appearance dissat-isfaction.
The?ndings relating to the moderating role of body apprecia-tion are novel and incremental to the body image literature.For women low on body appreciation,viewing models,compared to viewing control images,did not impact on the size of appearance-discrepancies but it did make appearance-discrepancies more salient.Interestingly,body appreciation also moderated the effects of media exposure among thin-ideal internalizing women.Pre-vious research has demonstrated,and this study con?rms,that women who internalize the thin-ideal are vulnerable to negative exposure effects.When women were high on thin-ideal internaliza-tion and low on body appreciation,viewing models was associated with larger and more salient appearance-discrepancies compared to the control group.However,when women high on thin-ideal internalization were also high on body appreciation,the negative exposure effects on appearance-discrepancy size disappeared and on appearance-discrepancy salience were reversed.This group of women reported less salient appearance-discrepancies after view-ing models compared to the control group.This suggests that women who buy into the thin-ideal but who also appreciate their bodies downplay the importance of appearance-discrepancies after viewing images of media models.
The inclusion of measures assessing both the size and the salience of appearance dissatisfaction has proved useful in this research.Findings relating to the extent of appearance dissatis-faction,as assessed through the size of appearance-discrepancies, replicate?ndings of previous research on media exposure.The moderating impact of internalization was only seen on this mea-sure of state body image.In contrast,body appreciation moderated the impact of media exposure on the importance women place on appearance-discrepancies.The results suggest that when women low on body appreciation view ultra-thin models they place greater importance on their appearance dissatisfaction.The role of body appreciation is particularly interesting when it is considered in combination with thin-ideal internalization,and it is striking that even when women buy into dominant appearance ideals,body appreciation seems to protect them from exposure to these ideals.
The?ndings are consistent with the conceptualization of posi-tive body image(i.e.,body appreciation)as guiding self-protective cognitive processing of media images to minimize the impact of environmental messages about appearance.Indeed,women who internalize the thin-ideal but who also appreciate their bodies place less importance on appearance-discrepancies after viewing models than control images.This may provide insight into one of the pro-tective psychological mechanisms used by women with positive body image.When confronted with idealized and arti?cial media images of female beauty,women with positive body image may actively downplay the importance of appearance relative to func-tional aspects of their bodies or other aspects of their self-identity. It would be informative for future research to examine salience rat-ing for other self-domains.In addition,the lack of exposure effects on appearance-discrepancy size for women high in both thin-ideal internalization and body appreciation suggests that these women are avoiding,or undoing,evaluative appearance comparisons with media models.It would also be informative to develop methods to directly assess women’s processing of ultra-thin media images.
Limitations
The sample size of the current study was relatively small and consisted of mainly White,young,undergraduate women.Clearly, these results also need replication with a larger more diverse sam-ple.In the current study,different products were used in the control condition and the model condition.Only women in the control con-dition saw advertisements for make-up and a handbag and only women in the model condition viewed advertisements for clothes and a soft drink.It is possible that the products may have had an impact on self-discrepancies.This is less concerning because pre-vious research has demonstrated that while women’s body image is strongly affected by the presence of models in advertising,it is not affected by viewing appearance-related products compared to household-related products(Birkeland et al.,2005).However, future research should match the products shown in each condi-tion.
Implications for Clinical Practice
Research has established a number of individual differences variables that make women vulnerable to negative societal mes-sages around appearance.The current?ndings add support to the theory that positive body image protects women from exter-nal appearance pressures.There is evidence that media literacy interventions that give women and girls information about the arti?cial nature of media images can prevent negative media expo-sure effects(Halliwell,Easun,&Harcourt,2011;Posavac,Posavac,& Weigel,2001;Yamamiya,Cash,Melnyk,Posavac,&Posavac,2005). The results of this study suggest that promoting body acceptance, a central aspect of body appreciation,should also protect women from negative societal messages about appearance.
In support of this assertion,body acceptance activities have been identi?ed as an effective component of interventions for body dissatisfaction and disordered eating(Stice,Shaw,&Marti,2007). Furthermore,a body image intervention incorporating body accep-tance has been shown to increase girls’resilience to thin-idealized media(Halliwell&Diedrichs,2013).To date,body image inter-ventions have been designed to target negative body image,and their effectiveness is measured through their impact on nega-tive body image.However,it may be that existing programs are also promoting positive body image and that this is strengthen-ing their impact.It would be useful to include measures of body appreciation in intervention evaluations to assess this question.In addition,the growing evidence that positive body image is central to psychological health suggests that interventions should specif-ically target body appreciation.Therapeutic techniques such as mindfulness that focus on self-acceptance and promoting psycho-logical well-being more broadly have been tailored to promote body appreciation(Stewart,2004).Mindfulness interventions have not yet been evaluated in relation to positive body image.There is emerging evidence that mindfulness can reduce negative body image and problematic eating behaviors(e.g.,Alberts,Thewissen, &Raes,2012).Future research should address whether mindful-ness techniques present a useful addition to existing body image interventions.
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新北师大九年级数学下册知识点总结
新北师大九年级数学下 册知识点总结 Document number【SA80SAB-SAA9SYT-SAATC-SA6UT-SA18】
新北师大版九年级数学下册知识点总结 第一章 直角三角形边的关系 一．锐角三角函数 1.正切： 定义：在Rt△ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ， 即的邻边 的对边A A A ∠∠=tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大；∠A 越大，梯子越陡，tanA 的值越大。 2.正弦．． ： 定义：在Rt△ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即 斜边 的对边A A ∠=sin ; 3.余弦： 定义：在Rt△ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即 斜边的邻边 A A ∠=cos ;
图1 锐角A 的正弦、余弦和正切都是∠A 的三角函数当锐角A 变化时，相应的正弦、余弦和正切之也随之变化。 二．特殊角的三角函数值 三．三角函数的计算 1. 仰角:当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． 3.规律：利用特殊角的三角函数值表，可以看出，(1)当角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 4.坡度：如图2，坡面与水平面的夹角叫做坡角坡角的正切称为坡度．．．．．．．．．．． (或坡比．． )。用字母i 表示，即A l h i tan == 5.方位角：从某点的指北方向按顺时针转到目标方向的水平角，叫做方位角．．． 。如图3，OA 、OB 、OC 的方位角分别为45°、135°、225°。 6.方向角：指北或指南方向线与目标方向线所成的小于90°的水平角，叫做方向角．．． 。如图4，OA 、OB 、OC 、OD 的方向角分别是；北偏东30°，南偏东45°(东南方向)、南偏西为60°，北偏西60°。 7.同角的三角函数间的关系： 30 o 45 o 60 o sin α cos α tan α 1 图2 h i=h:l l B
北师大版数学九年级下册：综合测试题
D C B A 30° 45° E D C B A o 北师大版九年级数学下册检测试题 一、选择题（每小题3分，共30分） 1.在△ABC 中，∠C ＝90°，a 、b 、c 分别是∠A 、∠B 、∠C 的对边.则有（ ） A.b ＝a tan A B.b ＝c sin A C.a ＝c cos B D.C ＝a sin A 2.如图，已知AB 是⊙O 的直径，?BC ＝?CD ＝?DE ，∠BOC ＝40°， 那么∠AOE ＝（ ） A.40° B. 60° C.60° D.120° 3.如图2，已知BD 是⊙O 的直径，⊙O 的弦AC ⊥BD 于点E ，若∠AOD=60°，则∠DBC 的度数为 （ ） A.30° B.40° C.50° D.60° 4.如图4，在直角坐标系中，圆O 的半径为1 2y x =-+与圆O 的位置关系是（ ） Ａ．相离 Ｂ．相交 Ｃ．相切 Ｄ．以上三种情形都有可能 5.二次函数y=ax 2+bx+c 与一次函数y=ax+c ，它们在同一直角坐标系中的图象大致是（ ） A ． B ． C ． D ． 6.如图，从热气球C 处测得地面A 、B 两点的俯角分别为30°、45°，如果此时热气球C 处的高度CD 为100米，点A 、D 、B 在同一直线上，则AB 两点的距离是（ ） A.200米 3米 3 3+1)米 7.如图，点A 、B 、C 在⊙O 上，∠ACB ＝30°，则sin ∠AOB 的值是（ ） A. 1 2 B.22 C.32 D. 33 8.已知点A (1，y 1)，B (－2，y 2)，C (－2，y 3)在函数y ＝12x 2－1 2 的图像上.则y 1、y 2、y 3的 大小关系是（ ） A.y 1＜y 2＜y 3 B.y 1＞y 2＞y 3 C.y 1＞y 3＞y 2 D.y 3＞y 1＞y 9.如图，点A ，B 的坐标分别为（1, 4）和（4, 4）,抛物线m x a y +-=2 )(顶点在线段AB 上运动，与x 轴交于C 、D 两点（C 在D 的左侧），点横坐标最小值为3-，则点D 的横坐标最大值为( ) A ．－3 B ．1 C ．5 D ．8 E D B C A O C B A O O 1 1 1- 1- y x 图4
北师大版九年级下册数学期末试卷
北师大版九年级下册数学期末试卷 一．选择题（共10小题） 1．下列式子错误的是（） A．cos40°=sin50°B．tan15°?tan75°=1C．sin225°+cos225°=1 D．sin60°=2sin30° 2．一个公共房门前的台阶高出地面1.2米，台阶拆除后，换成供轮椅行走的斜坡，数据如图所示，则下列关系或说法正确的是（） A．斜坡AB的坡度是10°B．斜坡AB的坡度是tan10° B．C．AC=1.2tan10°米D．AB=米 3．已知，在Rt△ABC中，∠C=90°，AB=，AC=1，那么∠A的正切tanA等于（）A．B．2 C． D． 4．函数y=k（x﹣k）与y=kx2，y=（k≠0），在同一坐标系上的图象正确的是（） A．B．C．D． 5．若抛物线y=x2﹣2x+3不动，将平面直角坐标系xOy先沿水平方向向右平移一个单位，再沿铅直方向向上平移三个单位，则原抛物线图象的解析式应变为（） A．y=（x﹣2）2+3 B．y=（x﹣2）2+5 C．y=x2﹣1 D．y=x2+4 6．若二次函数y=ax2﹣2ax+c的图象经过点（﹣1，0），则方程ax2﹣2ax+c=0的解为（） A．x 1=﹣3，x 2 =﹣1 B．x 1 =1，x 2 =3 C．x 1 =﹣1，x 2 =3 D．x 1 =﹣3，x 2 =1 7．如图所示，⊙O的半径为13，弦AB的长度是24，ON⊥AB，垂足为N，则ON=（）
A ．5 B ．7 C ．9 D ．11 8．如图，线段AB 是⊙O 的直径，弦CD ⊥AB ，∠CAB=40°，则∠ABD 与∠AOD 分别等于（ ） A ．40°，80° B ．50°，100° C ．50°，80° D ．40°，100° 9．已知⊙O 的半径OD 垂直于弦AB ，交AB 于点C ，连接AO 并延长交⊙O 于点E ，若AB=8，CD=2，则△BCE 的面积为（ ） A ．12 B ．15 C ．16 D ．18 10．二次函数y=ax 2+bx+c （a ≠0）的图象如图所示，下列结论：①b ＜0；②c ＞0；③a+c ＜b ；④b 2﹣4ac ＞0，其中正确的个数是（ ） A ．1 B ．2 C ．3 D ．4 二．填空题（共10小题） 11．在△ABC 中，∠C=90°，AB=13，BC=5，则sinA 的值是 ． 12．在将Rt △ABC 中，∠A=90°，∠C ：∠B=1：2，则sinB= ． 13．已知cos α=，则 的值等于 ． 14．已知抛物线y=ax 2﹣3x+c （a ≠0）经过点（﹣2，4），则4a+c ﹣1= ． 15．若二次函数y=2x 2﹣4x ﹣1的图象与x 轴交于A （x 1，0）、B （x 2，0）两点，则+ 的值为 ． 16．已知M 、N 两点关于y 轴对称，且点M 在双曲线 上，点N 在直线y=﹣x+3上， 设点M 坐标为（a ，b ），则y=﹣abx 2+（a+b ）x 的顶点坐标为 ． 17．若⊙O 的直径为2，OP=2，则点P 与⊙O 的位置关系是：点P 在⊙O ．
(完整版)新北师大九年级数学下册知识点总结
新北师大版九年级数学下册知识点总结 第一章直角三角形边的关系 一?锐角三角函数 1. 正切： 定义：在Rt △ ABC 中，锐角/A 的对边与邻边的比叫做/A 的正切，记作 tanA , ① tanA 是一个完整的符号，它表示/A 的正切，记号里习惯省去角的符号“/”； ② tanA 没有单位，它表示一个比值，即直角三角形中/A 的对边与邻边的比； ③ tanA 不表示"tan ”乘以"A ”； ④ 初中阶段，我们只学习直角三角形中，/A 是锐角的正切； ⑤ tanA 的值越大，梯子越陡，ZA 越大；ZA 越大，梯子越陡，tanA 的值越大。 2. 正弦： 定义：在Rt △ ABC 中，锐角/A 的对边与斜边的比叫做/A 的正弦，记作sinA ，即sin A A 的对边 ................................... """■ 斜边 3. 余弦： 定义：在Rt △ ABC 中，锐角/A 的邻边与斜边的比叫做/A 的余弦，记作cosA ，即cosA A 的邻边 .............................. ■■■■■ 斜边 之变化 三?三角函数的计算 1. 仰角：当从低处观测高处的目标时，视线与水平线所成的锐角称为 仰角 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为 俯角 值随着角度的增大（或减小）而增大（或减小）；余弦值随着角度的增大 < sin a< 1， 0< cos a< 1。 4. 坡度：如图2,坡面与水平面的夹角叫做坡角坡角的正切称为坡度 i tan A l 5. 方位角：从某点的指北方向按顺时针转到目标方向的水平角，叫做方位角...。如图3，OA OB OC 的方位角分别为 45 °、135 °、225 °。 6. 方向角：指北或指南方向线与目标方向线所成的小于 90°的水平角，叫做方向角.。如图4，OA 、 即 tanA A 的对边 A 的邻边 锐角A 的正弦、余弦和正切都是/A 的三角函数当锐角 A 变化时，相应的正弦、余弦和正切之也随 30 o 45 o 60 o sin a 1 亞 矗 2 2 2 cos a 旦 返 1 2 2 2 tan a 迴 3 1 3. 规律：禾U 用特殊角的三角函数值表，可以看岀, （1）当角度在0 °?90°间变化时，正弦值、正切 （或减小）而减小（或增大）。（2）0 （或坡比）。用字母i 表示，即 二?特殊角的三角函数值 图2
北师大版九年级数学下册全套教案
乂务教育基础课程初中教学资料 -- 第一章直角三角形的边角关系 § 1.1 从梯子的倾斜程度谈起（第一课时） 学习目标： 1. 经历探索直角三角形中边角关系的过程.理解正切的意义和与现实生活的联系. 2. 能够用tanA表示直角三角形中两边的比，表示生活中物体的倾斜程度、坡度等，外能够用正切进行简单的计算.学习重点：........ .... 1. 从现实情境中探索直角三角形的边角关系 2. 理解正切、倾斜程度、坡度的数学意义，密切数学与生活的联系学习难点： 理解正切的意义，并用它来表示两边的比. 学习方法： 引导一探索法. 学习过程： 一、生活中的数学问题： 1、你能比较两个梯子哪个更陡吗？你有哪些办法？ 2、生活问题数学化： ⑴如图：梯子 AB和EF哪个更陡？你是怎样判断的？ 哪个更陡？你是怎样判断的？ B 2m C F 2.5m D ≡EL-?P 二组第三组二、直角三角形的边与角的关系（如图，回答下列问题） ⑴Rt △ AB I Cl和Rt△ AB2C2有什么关系？ ⑵B I C l和BC L有什么关系？ AC1 AC 2 ⑶如果改变B2在梯子上的位置（如B3C3）呢？ ⑷由此你得出什么结论？ 三、例题： 例1、如图是甲，乙两个自动扶梯，哪一个自动扶梯比较陡 A C3C2 C]例2、在△ ABC中，∠ C=90°, BC=12cm AB=20cm 求tanA 和tan B 的值. E'l Sm R
1: 1.5的斜坡AD,求DB 的长.（结果保留根号） 五、课后练习: 1、 在 Rt △ ABC 中，∠ C=90 ,AB=3,BC=1,则 tanA= _______ 2、 在厶 ABC 中,AB=10,AC=8,BC=6,贝U tanA= ________ . 3、在厶 ABC 中,AB=AC=3,BC=4,则 tanC= ________ 四、随堂练习： 1如图，△ ABC 是等腰直角三角形，你能根据图中所给数据求出 tanC 吗? 2、如图，某人从山脚下的点 A 走了 200m 后到达山顶的点 B,已知点B 到山脚的垂直距离为 55m 求山的坡度?（结 3、若某人沿坡度i = 3: 4的斜坡前进10米，则他所在的位置比原来的位置 升高 ____________ 米. 4、菱形的两条对角线分别是 16和12.较长的一条对角线与菱形的一边的夹角为 tan θ = ______ . 5、如图，Rt △ ABC 是 一防洪堤背水坡的横截面图，斜坡 AB 的长为12 m ,它的坡角为45 ,为了提高该堤的 防洪能力，现将背水坡改造成坡比为 4、在 Rt △ ABC 中，∠C 是直角，∠A ∠ B ∠C 的对边分别是 a 、b 、c,且 a=24,c= 25, 求 tanA 、tanB 的值. 5、若三角形三边的比是 25:24:7,求最小角的正切值 5 6、如图，在菱形ABCc 中,AE ⊥BC 于E,EC=1,tanB= ,求菱形的边长和四 12 边形AECD 勺周长.
北师大版初三下册数学知识点总结
第一章 直角三角形边的关系 ※一. 正切： 定义：在Rt △ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．．，记作tanA ， 即的邻边 的对边 A A A ∠∠= tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大；∠A 越大，梯子越陡，tanA 的值越大。 ※二. 正弦．．： 定义：在Rt △ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即斜边 的对边 A A ∠= sin ; ※三. 余弦： 定义：在Rt △ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即斜边 的邻边 A A ∠= cos ; ※余切： 定义：在Rt △ABC 中，锐角∠A 的邻边与对边的比叫做∠A 的余切，记作cotA ，即的对边 的邻边 A A A ∠∠= cot ; ※一个锐角的正弦、余弦、正切、余切分别等于它的余角的余弦、正弦、余切、正切。 （通常我们称正弦、余弦互为余函数。同样，也称正切、余切互为余函数，可以概括为：一个锐角的三角函数等 于它的余角的余函数）用等式表 达：若∠A 为锐角，则 ①)90cos(sin A A ∠-?=； )90sin(cos A A ∠-?= ②)90cot(tan A A ∠-?=； )90tan(cot A A ∠-?=
图 1 图 3 图4 ※当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． ※当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． ※利用特殊角的三角函数值表，可以看出，(1)当角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值、余切值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 ※同角的三角函数间的关系： 倒数关系：tg α·ctg α=1。 ※在直角三角形中，除直角外，一共有五个元素，即三条边和二个锐角。由直角三角形中除直角外的已知元素，求出所有未知元素的过程，叫做解直角三角形。 ◎在△ABC 中，∠C 为直角，∠A 、∠B 、∠C 所对的边分别为a 、b 、c ，则有 (1)三边之间的关系：a 2+b 2=c 2； (2)两锐角的关系：∠A ＋∠B=90°； (3)边与角之间的关系： ;cot , tan , cos , sin a b A b a A c b A c a A ==== ;cot , tan , cos , sin b a B a b B c a B c b B ==== (4)面积公式:c ch ab 2 1 21S == ?(h c 为C 边上的高); (5)直角三角形的内切圆半径2c b a r -+= (6)直角三角形的外接圆半径c R 2 1 = ◎解直角三角形的几种基本类型列表如下： ◎解直角三角形的几种基本类型列表如下： 图2 h i=h:l B C
北师大版九年级数学下册各章知识点汇总
第一章 直角三角形的边角关系 1 锐角三角函数 2 30°，45°，60°角的三角函数值 3 三角函数的计算 4 解直角三角形 5 三角函数的应用 6 利用三角函数测高 ※一. 正切： 定义：在Rt △ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ，即的邻边 的对边 A A A ∠∠= tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan ”乘以“A ”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大； ∠A 越大，梯子越陡，tanA 的值越大。 ※二. 正弦．． ： 定义：在Rt △ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即 斜边 的对边 A A ∠= sin ; ※三. 余弦： 定义：在Rt △ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即 斜边 的邻边 A A ∠= cos ;
※余切： 定义：在Rt △ABC 中，锐角∠A 的邻边与对边的比叫做∠A 的余切，记作cotA ，即 的对边 的邻边 A A A ∠∠= cot ; ※一个锐角的正弦、余弦、正切、余切分别等于它的余角的余弦、正弦、余切、正切。 0o 30 o 45 o 60 o 90 o sin α 0 2 1 2 2 2 3 1 cos α 1 23 2 2 2 1 0 tan α 0 3 3 1 3 — cot α — 3 1 3 3 0 （通常我们称正弦、余弦互为余函数。同样，也称正切、余切互为余函数，可以概括为：一个锐角的三角函数等于它的余角的余函数）用等式表达：若∠A 为锐角，则 ①)90cos(sin A A ∠-?=； )90sin(cos A A ∠-?= ②)90cot(tan A A ∠-?=； )90tan(cot A A ∠-?= ※当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． ※当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． ※利用特殊角的三角函数值表，可以看出，(1)当 角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值、余切值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 ※同角的三角函数间的关系： 倒数关系：tg α·ctg α=1。
2020新版北师大版数学九年级下册教案(全)
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最新北师大版初三下册数学知识点
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新北师大九年级数学下册知识点总结 文件排版存档编号：[UYTR-OUPT28-KBNTL98-UYNN208]
新北师大版九年级数学下册知识点总结 第一章 直角三角形边的关系 一．锐角三角函数 1.正切： 定义：在Rt△ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ， 即的邻边 的对边A A A ∠∠=tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大；∠A 越大，梯子越陡，tanA 的值越大。 2.正弦．． ： 定义：在Rt△ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即斜边的对边 A A ∠=sin ; 3.余弦：
图1 定义：在Rt△ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即斜边的邻边 A A ∠=cos ; 锐角A 的正弦、余弦和正切都是∠A 的三角函数当锐角A 变化时，相应的正弦、余弦和正切之也随之变化。 二．特殊角的三角函数值 三．三角函数的计算 1. 仰角:当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． 3.规律：利用特殊角的三角函数值表，可以看出，(1)当角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 4.坡度：如图2，坡面与水平面的夹角叫做坡角坡角的正切称为坡度．．．．．．．．．．． (或坡比．．)。用字母i 表示，即A l h i tan == 5.方位角：从某点的指北方向按顺时针转到目标方向的水平角，叫做方位角．．．。如图3，OA 、OB 、OC 的方位角分别为45°、135°、225°。 30 o 45 o 60 o sin α cos α tan α 1 图2 h i=h: l B C
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北师大版九年级数学下册第一单元测试题 一、选择题(每题3分，共30分) 1．cos 30°的值为( )A.12 B.32 C.22 D.33 2．如图，已知Rt △BAC 中，∠C ＝90°，AC ＝4，tan A ＝12，则BC 的长是( ) A ．2 B ．8 C ．25 D ．4 5 (第2题)(第3题) 3．如图，在Rt △ABC 中，∠ACB ＝90°，CD ⊥AB 于D 点，已知AC ＝5，BC ＝2，那么 sin ∠ACD 等于( )A.53 B.23 C.253 D.52 4．若3tan (α＋10°)＝1，则锐角α的度数是( )A ．20° B ．30° C ．40° D ．50° 5．已知cos θ＝0.253 4，则锐角θ约等于( )A ．14.7° B ．14°7′ C ．75.3° D ．75°3′ 6．如图，某课外活动小组在测量旗杆高度的活动中，已测得仰角∠CAE ＝33°，AB ＝a ，BD ＝b ，则下列求旗杆CD 长的式子中正确的是( ) A ．CD ＝b sin 33°＋a B ．CD ＝b cos 33°＋a C ．C D ＝b tan 33°＋a D ．CD ＝b tan 33°＋a (第6题)(第7题) 7．如图，在网格中，小正方形的边长均为1，点A ，B ，C 都在格点上，则∠ABC 的正切值是 ( )A ．2 B .255 C.55 D.12 8．在△ABC 中，∠A ＝30°，∠B ＝45°，AB ＝2(1＋3)，则BC 等于( ) A ．2 B. 6 C ．2 2 D ．1＋ 3 9．如图，在高楼前D 点测得楼顶的仰角为30°，向高楼前进60 m 到C 点，又测得仰角为45°，
北师大版数学九年级下册教案-最新(全)
第一章直角三角形的边角关系 第1课时 §1.1.1 锐角三角函数 教学目标 1、经历探索直角三角形中边角关系的过程 2、理解锐角三角函数（正切、正弦、余弦）的意义，并能够举例说明 3、能够运用三角函数表示直角三角形中两边的比 4、能够根据直角三角形中的边角关系，进行简单的计算 教学重点和难点 重点：理解正切函数的定义 难点：理解正切函数的定义 教学过程设计 ?从学生原有的认知结构提出问题 直角三角形是特殊的三角形，无论是边，还是角，它都有其它三角形所没有的性质。这一章，我们继续学习直角三角形的边角关系。 ?师生共同研究形成概念 1、梯子的倾斜程度 在很多建筑物里，为了达到美观等目的，往往都有部分设计成倾斜的。这就涉及到倾斜角的问题。用倾斜角刻画倾斜程度是非常自然的。但在很多实现问题中，人们无法测得倾斜角，这时通常采用一个比值来刻画倾斜程度，这个比值就是我们这节课所要学习的——倾斜角的正切。 1）（重点讲解）如果梯子的长度不变，那么墙高与地面的比值越大，则梯子越陡；2）如果墙的高度不变，那么底边与梯子的长度的比值越小，则梯子越陡； 3）如果底边的长度相同，那么墙的高与梯子的高的比值越大，则梯子越陡； 通过对以上问题的讨论，引导学生总结刻画梯子倾斜程度的几种方法，以便为后面引入正切、正弦、余弦的概念奠定基础。 2、想一想（比值不变） ☆想一想书本P 2 想一想 通过对前面的问题的讨论，学生已经知道可以用倾斜角的对边与邻边之比来刻画梯子的倾斜程度。当倾斜角确定时，其对边与邻边的比值随之确定。这一比值只与倾斜角的大小有关，而与直角三角形的大小无关。
3、 正切函数 （1） 明确各边的名称 （2） 的邻边 的对边A A A ∠∠=tan （3） 明确要求：1）必须是直角三角形；2）是∠A 的对边与∠A 的邻边的比值。 ☆ 巩固练习 a 、 如图，在△ACB 中，∠C = 90°， 1） tanA = ；tanB = ； 2） 若AC = 4，BC = 3，则tanA = ；3） 若AC = 8，AB = 10，则tanA = ；tanB = ；b 、 如图，在△ACB 中，tanA = 。（不是直角三角形） （4） tanA 的值越大，梯子越陡 4、 讲解例题 例1 图中表示甲、乙两个自动扶梯，哪一个自动扶梯比较陡？ 分析：通过计算正切值判断梯子的倾斜程度。这是上述结论的直接应用。 例2 如图，在△ACB 中，∠C = 90°，AC = 6，4 3 tan = B ，求B C 、AB 的长。 分析：通过正切函数求直角三角形其它边的长。 ? 随堂练习 5、书本 P 4 随堂练习 ? 小结 正切函数的定义。 ? 作业 书本 P4 习题1.1 1、2、4。 B A B C ∠A 的对边 ∠A 的邻边斜边 8m α5m 5m β 13m A B C
北师大版九年级数学下学期数学教学计划(最新整理)
九年级数学第二学期教学计划 一、学情分析： 九年级(2)班成绩一般，两极分化严重，经过上一学期的努力，很多学生在学习风气上有了较大的改变，学习积极性有所提高，也有不少学生自知能力较差，特别是到了最后一学期，最自己要求不严，甚至自暴自弃，这些都需要针对不同情况采取相应的措施，耐心教育，此外，面临中考阶段对学生要有总体的掌握，使之考出好成绩。 二、本册教材教学目标： 1、情感目标及价值观： 通过学习交流、合作、讨论的方式，积极探索，激发学生的学习兴趣，改进学生的学习方式，提高学习质量，逐步形成正确的教学价值观，使学生的情感得到发展。 2、知识与技能 理解点、直线、圆与圆的位置关系，弧长和扇形的面积，圆锥的侧面展开图，掌握圆的切线及与圆有关的角等概念和计算。教育学生掌握基础知识与基本技能，培养学生的逻辑思维能力、运算能力、空间观念和解决简单实际问题的能力，使学生逐步学会正确、合理的进行运算，逐步学会观察分析、综合、抽象、概括。会用归纳演绎、类比进行简单的推理，提高学生学习数学的兴趣，逐步培养学生具有良好的学习习惯，实事求是的态度，掌握初中数学教材、数学学科“基本要求”的知识点。 3、过程与方法： 经历探索过程，让学生进一步体会数学来源与实践，又反应用于实践，通过探索、学习，使学生逐步学会正确、合理的进行运算，逐步学会观察、
分析、综合、抽象、会用归纳、演绎、类比进行简单的推理，围绕初中数学教材、数学学科“基本要求”进行知识梳理，围绕初中数学主要内容进行专题复习，适时地进行分层教学，面向全体学生、培养学生、发展全体学生。 三、本册教材分析 本学期的内容只剩一章：圆。 圆这一章的主要内容是圆的定义和性质，点、直线、圆与圆的位置关系，圆的切线，弧长和扇形的面积。本章设涉及的概念、定理较多，应弄清来龙去脉，准确理解和掌握概念和定理。垂径定理及推论、圆的切线的判定定理和性质定理是本章的重点。垂径定理、圆周角定理的证明、运用与圆有关的性质解决实际问题，是本章的难点。 除了这一章，还要复习初中数学教材其他的内容。 四、教学重难点 重点： 圆这章中垂径定理及推论、圆的切线的判定定理和性质定理是本章 的重点。 难点： 垂径定理、圆周角定理的证明、运用与圆有关的性质解决实际问题。 五、教学中要采取的措施： 1、认真学习钻研新课标，通盘熟悉初中数学教材及教学目标，认真备好每一堂课，精心制作总复习计划。 2、认真上好每一堂课，抓住关键，分散难点，突出重点，在培养能力上下功夫。 3、重视课后反思，及时将每一节课的得失记录下来，不断的积累教学经
新北师大版九年级数学下册知识点汇总
新北师大版九年级数学下册知识点汇总 第一章 直角三角形边的关系 一．锐角三角函数 1.正切： 定义：在Rt△ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ， 即的邻边 的对边A A A ∠∠=tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大；∠A 越大，梯子越陡，tanA 的值越大。 2.正弦．． ： 定义：在Rt△ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即斜边 的对边A A ∠=sin ; 3.余弦： 定义：在Rt△ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即斜边的邻边A A ∠=cos ; 锐角A 的正弦、余弦和正切都是∠A 的三角函数当锐角A 变化时，相应的正弦、余弦和正切之也随之变化。 二．特殊角的三角函数值
图 1 图 3 图4 三．三角函数的计算 1. 仰角:当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． 3.规律：利用特殊角的三角函数值表，可以看出，(1)当角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 4.坡度：如图2，坡面与水平面的夹角叫做坡角坡角的正切称为坡度．．．．．．．．．．． (或坡比．． )。用字母i 表示，即A l h i tan == 5.方位角：从某点的指北方向按顺时针转到目标方向的水平角，叫做方位角．．． 。如图3，OA 、OB 、OC 的方位角分别为45°、135°、225°。 6.方向角：指北或指南方向线与目标方向线所成的小于90°的水平角，叫做方向角．．． 。如图4，OA 、OB 、OC 、OD 的方向角分别是；北偏东30°，南偏东45°(东南方向)、南偏西为60°，北偏西60°。 7.同角的三角函数间的关系： ①互余关系sinA=cos(90°－A)、cosA=sin(90°－A) ②平方关系：③商数关系： 8.解直角三角形：在直角三角形中，除直角外，一共有五个元素，即三条边和二个锐角。由直 图2 h i=h:l A B C
(完整版)北师大九年级数学下册知识点汇总
图1 北师大版初中数学定理知识点汇总[九年级(下册) 第一章 直角三角形边的关系 ※一. 正切： 定义：在Rt △ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ，即的邻边 的对边 A A A ∠∠=tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大； ∠A 越大，梯子越陡，tanA 的值越大。 ※二. 正弦．．： 定义：在Rt △ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即斜边 的对边 A A ∠=sin ; ※三. 余弦： 定义：在Rt △ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即斜边的邻边 A A ∠=cos ; ※余切： 定义：在Rt △ABC 中，锐角∠A 的邻边与对边的比叫做∠A 的余切，记作cotA ，即的对边的邻边 A A A ∠∠= cot ; ※一个锐角的正弦、余弦、正切、余切分别等于它的余角的余弦、正弦、余切、正切。 （通常我们称正弦、余弦互为余函数。同样，也称正切、余切互为余函数，可以概括为：一个锐角的三角函数等于它的余角的余函数）用等式表达：若∠A 为锐角，则 ①)90cos(sin A A ∠-?=； )90sin(cos A A ∠-?= ②)90cot(tan A A ∠-?=； )90tan(cot A A ∠-?= ※当从低处观测高处的目标时，视线与水平线 所成的锐角称为仰角．． ※当从高处观测低处的目标时，视线与水平线所成 的锐角称为俯角．． ※利用特殊角的三角函数值表，可以看出，(1)当 角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值、余切值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 ※同角的三角函数间的关系： 倒数关系：tg α·ctg α=1。 ※在直角三角形中，除直角外，一共有五个元素，即三条边和二个锐角。由直角三角形中除直角外的已知元素，求出 0o 30 o 45 o 60 o 90 o sin α 0 2 1 22 2 3 1 cos α 1 23 2 2 2 1 0 tan α 0 3 3 1 3 — cot α — 3 1 3 3 0
北师大九年级数学下册知识点总结
图 1 图 3 图4 新北师大版九年级数学下册知识点总结 第一章 直角三角形边的关系 一．锐角三角函数 1.正切： 定义：在Rt△ABC 中，锐角∠A 的对边与邻边的比叫做∠A 的正切．． ，记作tanA ， 即的邻边 的对边A A A ∠∠=tan ; ①tanA 是一个完整的符号，它表示∠A 的正切，记号里习惯省去角的符号“∠”； ②tanA 没有单位，它表示一个比值，即直角三角形中∠A 的对边与邻边的比； ③tanA 不表示“tan”乘以“A”； ④初中阶段，我们只学习直角三角形中，∠A 是锐角的正切； ⑤tanA 的值越大，梯子越陡，∠A 越大；∠A 越大，梯子越陡，tanA 的值越大。 2.正弦．． ： 定义：在Rt△ABC 中，锐角∠A 的对边与斜边的比叫做∠A 的正弦，记作sinA ，即斜边 的对边A A ∠=sin ; 3.余弦： 定义：在Rt△ABC 中，锐角∠A 的邻边与斜边的比叫做∠A 的余弦，记作cosA ，即斜边 的邻边A A ∠=cos ; 锐角A 的正弦、余弦和正切都是∠A 的三角函数当锐角A 变化时，相应的正弦、余弦和正切之也随之变化。 二．特殊角的三角函数值 三．三角函数的计算 1. 仰角:当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角．． 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角．． 3.规律：利用特殊角的三角函数值表，可以看出，(1)当角度在0°～90°间变化时，正弦值、正切值随着角度的增大(或减小)而增大(或减小)；余弦值随着角度的增大(或减小)而减小(或增大)。(2)0≤sin α≤1，0≤cos α≤1。 4.坡度：如图2，坡面与水平面的夹角叫做坡角坡角的正切称为坡度．．．．．．．．．．． (或坡比．． )。用字母i 表示，即A l h i tan == 5.方位角：从某点的指北方向按顺时针转到目标方向的水平角，叫做方位角．．． 。如图3，OA 、OB 、OC 的方位角分别为45°、135°、225°。 6.方向角：指北或指南方向线与目标方向线所成的小于90°的水平角，叫做方向角．．． 。如图4，OA 、OB 、OC 、OD 的方向角分别是；北偏东30°，南偏东45°(东南方向)、南偏西为60°，北偏西60°。 7.同角的三角函数间的关系： ①互余关系sinA=cos(90°－A)、cosA=sin(90°－A ) 30 o 45 o 60 o sin α cos α tan α 1 图2 h i=h:l B C
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