Problem 142 - Precalculus with Limits

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.

Step-by-Step Solution

Answer

The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \)can be derived from the definition of sine and cosine, and the identities \( 1 + \tan^2 \theta = \sec^2 \theta \) and \( 1 + \cot^2 \theta = \csc^2 \theta \) can be derived from this Pythagorean identity. Remembering these can be made easier by relating them to the unit circle and understanding the relationships between the trigonometric functions.

Step 1: Derive the Pythagorean identity

Using a right-angle triangle definition, we have \(\sin \theta = \frac{opposite}{hypotenuse}\) and \(\cos \theta = \frac{adjacent}{hypotenuse}\). Squaring both sides and adding them, we get \(\sin^2 \theta + \cos^2 \theta = \frac{opposite^2}{hypotenuse^2} + \frac{adjacent^2}{hypotenuse^2}\). Using the Pythagorean Theorem, we get \(\sin^2 \theta + \cos^2 \theta = 1 \).

Step 2: Derive the identity \(1 + \tan^2 \theta = \sec^2 \theta\)

We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta}\), so \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\). On the other hand, \( \sec \theta = \frac{1}{\cos \theta}\), so \( \sec^2 \theta = \frac{1}{\cos^2 \theta}\). Replacing \( \sec^2 \theta \) in the Pythagorean identity, we get \(1 + \tan^2 \theta = \sec^2 \theta\).

Step 3: Derive the identity \(1 + \cot^2 \theta = \csc^2 \theta\)

We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta}\), so \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}\). On the other hand, \( \csc \theta = \frac{1}{\sin \theta}\), so \( \csc^2 \theta = \frac{1}{\sin^2 \theta}\). Replacing \( \csc^2 \theta \) in the Pythagorean identity, we get \(1 + \cot^2 \theta = \csc^2 \theta\).

Step 4: Discuss how to remember these identities

To remember these identities, it may be helpful relating them back to the unit circle. Additionally, understanding the relationships between the various trigonometric functions can also aid in memorizing these identities.