Triangle Method

https://www.youtube.com/watch?v=r82erwtu5vA Changing an angle theta to be the result of a function.

Examples

General Example

Rewrite $\tan ({\sec }^{-1}(x))$ using the triangle method.

Solution

1. Let $\theta ={\sec }^{-1}(x)$
2. Note that $\theta \in [0,\frac{\pi }{2})\cup [\frac{\pi }{2},\pi ]$
3. Then note that $\cos (\theta )=\frac{1}{x}$
4. Draw a triangle using this information
5. Case 1: $\theta \in [0,\frac{\pi }{2})$
   1. Then $\tan (\theta )=\frac{\sqrt{x^2-1}}{1}=\sqrt{x^2-1}$
   2. Thus $x>1$
6. Case 2: $\theta \in (\frac{\pi }{2},\pi ]$
   1. Then $\tan (\theta )=\frac{-\sqrt{x^2-1}}{1}=-\sqrt{x^2-1}$
   2. Thus $x\le 1$

Specific Example

Drawing 2024-12-15 16.17.12.excalidraw