Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}\right)$$

Step-by-Step Solution

Answer

The exact real number value is \( \frac{63}{65} \).

Step 1: Understand the Problem

The expression \( \cos \left( \sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}\right) \) involves finding the cosine of a difference of angles. Each angle is defined in terms of sine and cosine inverses.

Step 2: Identify the Angles

Let \( x = \sin^{-1} \frac{3}{5} \) and \( y = \cos^{-1} \frac{12}{13} \). This means \( \sin x = \frac{3}{5} \) and \( \cos y = \frac{12}{13} \).

Step 3: Find the Cosines of the Angles

Using the identity \( \sin^2 x + \cos^2 x = 1 \), find \( \cos x \). \[\cos x = \sqrt{1 - \left( \frac{3}{5} \right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}\] and for \( y \), \( \sin^2 y + \cos^2 y = 1 \). \[ \sin y = \sqrt{1 - \left( \frac{12}{13} \right)^2} = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13} \]

Step 4: Use the Cosine of Angle Difference Identity

Use the identity \( \cos(x-y) = \cos x \cos y + \sin x \sin y \). Substitute the values found: \[ \cos(x-y) = \frac{4}{5} \cdot \frac{12}{13} + \frac{3}{5} \cdot \frac{5}{13} \]

Step 5: Calculate the Expression

Perform the multiplications and add the results: \[ \cos(x-y) = \frac{4 \times 12}{5 \times 13} + \frac{3 \times 5}{5 \times 13} = \frac{48}{65} + \frac{15}{65} = \frac{63}{65} \]

Step 6: Write the Exact Real Number Value

The exact real number value of the expression is \( \frac{63}{65} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the standard trigonometric functions like sine, cosine, and tangent. They help us find an angle when the value of the trigonometric function is given. For example, \( \sin^{-1} \) or arcsin finds the angle whose sine value is a specific number. Similarly, \( \cos^{-1} \) or arccos finds the angle whose cosine value is given. These functions are especially useful in trigonometry when dealing with problems involving right triangles or unit circles.

In this exercise, we're looking at two inverse trigonometric functions:

• \( \sin^{-1} \frac{3}{5} \) gives angle \( x \) whose sine is \( \frac{3}{5} \).
• \( \cos^{-1} \frac{12}{13} \) gives angle \( y \) whose cosine is \( \frac{12}{13} \).

Once the angles are determined using these functions, they allow us to further solve the trigonometric identities like cosine of a difference in angles.

Angle Difference Identity

The angle difference identity is a powerful tool in trigonometry that allows us to simplify expressions involving the cosine and sine of angle differences. Specifically, the identity for cosine is expressed as:

• \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]

This identity is used in different areas including geometry, calculus, and physics, to help in solving various problems involving angular transformations. It simplifies calculating the cosine of the difference of two angles by breaking it down into simpler computations of the sines and cosines of individual angles.

In the given problem, this identity is applied to find the cosine of the angle difference \( \cos(\sin^{-1} \frac{3}{5} - \cos^{-1} \frac{12}{13}) \), by substituting the values of \( \cos x \), \( \cos y \), \( \sin x \), and \( \sin y \) derived from the previous steps.

Exact Values Computation

Computing the exact values of trigonometric expressions means finding a precise and simplified numerical result without resorting to approximations or calculators. This requires a thorough understanding of how to manipulate algebraic expressions and apply trigonometric identities as we have in this exercise.

Here, we compute the exact value of \( \cos(x-y) \) by performing separate calculation steps:

• Calculate \( \cos x \) and \( \sin y \) based on the derived expressions from earlier steps using Pythagoras' identity.
• Substitute these values into the cosine angle difference identity.
• Perform the arithmetic multiplication and add the terms: \( \frac{4}{5} \times \frac{12}{13} + \frac{3}{5} \times \frac{5}{13} = \frac{63}{65} \).

This accurate computation provides the exact real number value of the expression \( \frac{63}{65} \), demonstrating the effectiveness of systematic problem-solving in trigonometry.