Problem 54 - Precalculus Essentials

Question

Use a sketch to find the exact value of each expression. $$\cos \left(\sin ^{-1} \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \cos(\sin^{-1}\frac{1}{2}) \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Compute inverse sine

First, calculate \( \sin^{-1}\frac{1}{2} \). We look for an angle whose sine is 1/2. This angle is usually \( \frac{\pi}{6} \) or \( 30^\circ \). Therefore, \( \sin^{-1}\frac{1}{2} = \frac{\pi}{6} \).

Step 2: Compute cosine of the result

Afterwards, calculate \( \cos(\frac{\pi}{6}) \). The cosine of \( \frac{\pi}{6} \) or \( 30^\circ \) is \( \frac{\sqrt{3}}{2} \) . Therefore, \( \cos(\sin^{-1}\frac{1}{2}) = \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that relate various trigonometric functions to one another. These identities are often used to simplify expressions or solve trigonometric equations. A basic understanding of these identities can help you manipulate and understand more complex trigonometric expressions.
For instance, one of the most used fundamental identities is the Pythagorean Identity, which is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity tells us that the square of the sine of an angle plus the square of the cosine of that same angle is always equal to 1. By remembering and using these identities, trigonometry becomes easier to master. Insights from identities help while dealing with inverse trigonometric functions and graph interpretations too.

Cosine Function

The cosine function, one of the primary trigonometric functions, relates the angle of the right triangle to the adjacent side over the hypotenuse. It is an even function, which means that it exhibits symmetry, specifically

• \( \cos(-\theta) = \cos(\theta) \).

This function is periodic with a period of \( 2\pi \), meaning that its shape repeats every \( 2\pi \) radians.
Understanding the characteristics of the cosine function is essential for evaluating expressions involving it. When it comes to inverse sine, for instance, understanding the cosine of angles derived from inverse trigonometric functions requires familiarity. The cosine function's values are calculated by knowing specific angles and their trigonometric values, which helps in solving problems like finding the exact trigonometric values.

Exact Trigonometric Values

Exact trigonometric values refer to the precise value of trigonometric functions at specific standard angles, such as

• \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and so on.

These angles are very useful because their trigonometric values are commonly memorized and used frequently in trigonometric calculations. For example,

• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)

By using these exact values, trigonometric calculations become quicker and more straightforward. This is particularly advantageous when building upon other methods, such as inverse trigonometric functions, where understanding these values leads to the final solution efficiently. Furthermore, these measures help provide a deep comprehension of the trigonometric unit circle and the associated relationships between different trigonometric functions.