Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.4 Trigonometric Functions - Exercises - Page 30: 29

Answer

Picture (A). For the point A: $$\sin\alpha=0.918$$ $$\cos\alpha=0.3965$$ $$\tan\alpha\approx 2.315$$ For the point B: $$\sin\angle B =0.3965, \cos\angle B = - 0.918, $$ $$\tan\angle B = -\dfrac{0.3965}{0.918}\approx - 0.432$$ For the point C: $$\sin\angle C =-0.918, \cos\angle C = - 0.3965, $$ $$\tan\angle C = \dfrac{0.918}{0.3965}\approx 2.315$$ For the point D: $$\sin\angle D =-0.3965, \cos\angle D = 0.918, $$ $$\tan\angle D = -\dfrac{0.3965}{0.918}\approx - 0.432$$ Picture (B): For the point A: $$\sin\alpha=0.918$$ $$\cos\alpha=0.3965$$ $$\tan\alpha\approx 2.315$$ For the point B: $$\sin\angle B =0.918, \cos\angle B = - 0.3965, $$ $$\tan\angle B = -\dfrac{0.3965}{0.918}\approx - 2.315$$ For the point C $$\sin\angle C =-0.918, \cos\angle C = - 0.3965, $$ $$\tan\angle C = \dfrac{0.918}{0.3965}\approx 2.315$$ For the point D $$\sin\angle D =-0.918, \cos\angle D = 0.3965, $$ $$\tan\angle D = -\dfrac{0.918}{0.3965}\approx -2.315$$

Work Step by Step

Let's start with pic. (A) For the point A: $$\sin\alpha=y=0.918$$ $$\cos\alpha=x=0.3965$$ $$\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{y}{x}=\dfrac{0.918}{0.3965}\approx 2.315$$ For the point B the angle (formally call it $\angle B$) is obvious $\dfrac{\pi}{2} + \alpha$. For the point C the angle ($\angle C$) is $\pi + \alpha.$ And for the point D ($\angle D$) is $\dfrac{3\pi}{2}+\alpha.$ We can use the equalities: $$\sin(\pi/2+\alpha)=\cos\alpha, \cos(\pi/2+\alpha)=-\sin\alpha$$ $$\tan(\pi/2+\alpha)=\dfrac{\sin(\pi/2+\alpha)}{ \cos(\pi/2+\alpha)}=-\dfrac{\cos\alpha}{\sin\alpha}$$ So $$\sin\angle B =0.3965, \cos\angle B = - 0.918, $$ $$\tan\angle B = -\dfrac{0.3965}{0.918}\approx - 0.432$$ For the point C: $$\sin(\pi+\alpha)=-\sin\alpha, \cos(\pi+\alpha)=-\cos\alpha$$ $$\tan(\pi+\alpha)=\dfrac{\sin(\pi+\alpha)}{ \cos(\pi+\alpha)}=\dfrac{\cos\alpha}{\sin\alpha}$$ And $$\sin\angle C =-0.918, \cos\angle C = - 0.3965, $$ $$\tan\angle C = \dfrac{0.918}{0.3965}\approx 2.315$$ For the point D $$\sin(3\pi/2+\alpha)=-\cos\alpha, \cos(3\pi/2+\alpha)=\sin\alpha$$ $$\tan(3\pi/2+\alpha)=\dfrac{\sin(3\pi/2+\alpha)}{ \cos(3\pi/2+\alpha)}=-\dfrac{\cos\alpha}{\sin\alpha}$$ And $$\sin\angle D =-0.3965, \cos\angle D = 0.918, $$ $$\tan\angle D = -\dfrac{0.3965}{0.918}\approx - 0.432$$ Now let's deal with picture (B). Let's act in a similar way. For the point A: $$\sin\alpha=y=0.918$$ $$\cos\alpha=x=0.3965$$ $$\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{y}{x}=\dfrac{0.918}{0.3965}\approx 2.315$$ For the point B the angle (formally call it $\angle B$) is obvious $\pi - \alpha$. For the point C the angle ($\angle C$) is $\pi + \alpha.$ And for the point D ($\angle D$) is $2\pi-\alpha.$ We can use the equalities: $$\sin(\pi-\alpha)=\sin\alpha, \cos(\pi-\alpha)=-\cos\alpha$$ $$\tan(\pi-\alpha)=\dfrac{\sin(\pi-\alpha)}{ \cos(\pi-\alpha)}=-\dfrac{\sin\alpha}{\cos\alpha}$$ So $$\sin\angle B =0.918, \cos\angle B = - 0.3965, $$ $$\tan\angle B = -\dfrac{0.3965}{0.918}\approx - 2.315$$ For the point C: $$\sin(\pi+\alpha)=-\sin\alpha, \cos(\pi+\alpha)=-\cos\alpha$$ $$\tan(\pi+\alpha)=\dfrac{\sin(\pi+\alpha)}{ \cos(\pi+\alpha)}=\dfrac{\cos\alpha}{\sin\alpha}$$ And $$\sin\angle C =-0.918, \cos\angle C = - 0.3965, $$ $$\tan\angle C = \dfrac{0.918}{0.3965}\approx 2.315$$ For the point D $$\sin(2\pi-\alpha)=-\sin\alpha, \cos(2\pi-\alpha)=\cos\alpha$$ $$\tan(2\pi-\alpha)=\dfrac{\sin(2\pi-\alpha)}{ \cos(2\pi-\alpha)}=-\dfrac{\sin\alpha}{\cos\alpha}$$ And $$\sin\angle D =-0.918, \cos\angle D = 0.3965, $$ $$\tan\angle D = -\dfrac{0.918}{0.3965}\approx -2.315$$
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