91Ó°ÊÓ

Inverse trigonometric functions are used to find the angle when a specific trigonometric ratio is known. For instance, with \(\sin^{-1}\), we determine the angle whose sine is a given value. When you see \(\sin^{-1} \frac{3}{4}\), it asks which angle within the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\) has a sine of \(\frac{3}{4}\).
\(\cos^{-1}\), on the other hand, deals with cosine values, finding angles within \([0, \pi]\). If \(\cos^{-1} \frac{1}{3}\) is given, we're asking which angle has cosine \(\frac{1}{3}\).
Inverse trigonometric functions are commonly used in conjunction with identities to solve problems, helping to shift between different types of trigonometric expressions smoothly.

The tangent of a difference formula comes in handy when you encounter terms like \(\tan(a-b)\). This formula states: \[\tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b}\] This formula simplifies the calculation of the tangent for expressions involving the difference between two angles. For example, in our exercise, \(a = \sin^{-1} \frac{3}{4}\) and \(b = \cos^{-1} \frac{1}{3}\).
First, find \(\tan a\) and \(\tan b\) by calculating \(\tan a = \frac{\sin a}{\cos a}\) and \(\tan b = \frac{\sin b}{\cos b}\). Then, plug these values into the formula.
This method avoids directly calculating the angles and is particularly useful in exact trigonometric value problems.

The Pythagorean identity is a key tool in trigonometry, which relates the square of sine and cosine functions: \[\cos^2 \theta + \sin^2 \theta = 1\] This identity is useful when you have either \(\sin \theta\) or \(\cos \theta\) and need to find the other. In our solution, knowing \(\sin \theta = \frac{3}{4}\), we used this identity to determine \(\cos \theta\) by rearranging it: \[\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{3}{4}\right)^2} = \frac{\sqrt{7}}{4}\] Likewise, knowing \(\cos \phi = \frac{1}{3}\), we found \(\sin \phi\) as: \[\sin \phi = \sqrt{1 - \cos^2 \phi} = \frac{2\sqrt{2}}{3}\] Use this identity whenever trigonometric expressions involving sine and cosine need simplifying or solving.