Question

(a) Use the definitions of sine and cosine to derive the Pythagorean identity $ \sin^2 \theta + \cos^2 \theta = 1 $. (b) Use the Pythagorean identity $ \sin^2 \theta + cos^2 \theta = 1 $ to derive the other Pythagorean identities, $ 1 + \tan^2 \theta = sec^2 \theta $ and $ 1 + \cot^2 \theta = \csc^2 \theta $. Discuss how to remember these identities and other fundamental identities.

Name: Problem 142, Chapter 5: Precalculus with Limits
Uploaded: 2020-07-17T03:28:16-08:00
Duration: 2 min 51 s
Channel: Heather Zimmers
Description: (a) Use the definitions of sine and cosine to derive the Pythagorean identity sin^2 θ+ cos^2 θ= 1 . (b) Use the Pythagorean identity sin^2 θ+ cos^2 θ= 1 to derive the other Pythagorean identities, 1 + tan^2 θ= sec^2 θand 1 + cot^2 θ= csc^2 θ. Discuss how to remember these identities and other fundamental identities.

   (a)  Use  the  definitions  of  sine  and  cosine  to  derive the  Pythagorean  identity $ \sin^2 \theta + \cos^2 \theta = 1 $.

(b)  Use  the  Pythagorean  identity $ \sin^2 \theta + cos^2 \theta = 1 $ to  derive  the  other  Pythagorean  identities, $ 1 + \tan^2 \theta = sec^2 \theta $ and $ 1 + \cot^2 \theta = \csc^2 \theta $. Discuss  how  to  remember  these  identities  and other  fundamental  identities.

Precalculus with Limits

Ron Larson 2nd Edition

Chapter 5, Problem 142

Instant Answer

Solved by Expert Heather Zimmers

07/17/2020

Step 1

If we have a right triangle with an angle $\theta$, the side opposite to $\theta$ is $y$, the side adjacent to $\theta$ is $x$, and the hypotenuse is $r$. Then, $\sin \theta = \frac{y}{r}$ and $\cos \theta = \frac{x}{r}$. If we isolate $y$ and $x$, we get $y = r Show more…

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(a) Use the definitions of sine and cosine to derive the Pythagorean identity $ \sin^2 \theta + \cos^2 \theta = 1 $. (b) Use the Pythagorean identity $ \sin^2 \theta + cos^2 \theta = 1 $ to derive the other Pythagorean identities, $ 1 + \tan^2 \theta = sec^2 \theta $ and $ 1 + \cot^2 \theta = \csc^2 \theta $. Discuss how to remember these identities and other fundamental identities.