Problem 94 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The exact value of the expression is \(-\frac{16}{65}\).

Step 1: Understanding the Problem

We need to find the exact value of the expression \( \cos \left( \sin^{-1} \frac{3}{5} + \cos^{-1} \frac{5}{13} \right) \). This involves using trigonometric identities and inverse trigonometric functions.

Step 2: Define the Angles Using Inverse Functions

Let \( \theta = \sin^{-1} \frac{3}{5} \) and \( \beta = \cos^{-1} \frac{5}{13} \). This implies that \( \sin \theta = \frac{3}{5} \) and \( \cos \beta = \frac{5}{13} \). We need to find \( \cos(\theta + \beta) \).

Step 3: Use Trigonometric Identities

We will use the identity \( \cos(\theta + \beta) = \cos \theta \cos \beta - \sin \theta \sin \beta \). We already have \( \cos \beta = \frac{5}{13} \) and \( \sin \theta = \frac{3}{5} \). We need to find \( \cos \theta \) and \( \sin \beta \).

Step 4: Find Remaining Trigonometric Values

For \( \sin \theta = \frac{3}{5} \), using \( \cos^2 \theta = 1 - \sin^2 \theta \), we get \( \cos \theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5} \). For \( \cos \beta = \frac{5}{13} \), using \( \sin^2 \beta = 1 - \cos^2 \beta \), we get \( \sin \beta = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13} \).

Step 5: Calculate \( \cos(\theta + \beta) \)

Substitute our known values into the identity: \( \cos(\theta + \beta) = \frac{4}{5} \cdot \frac{5}{13} - \frac{3}{5} \cdot \frac{12}{13} \). This simplifies to \( \cos(\theta + \beta) = \frac{20}{65} - \frac{36}{65} = \frac{-16}{65} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions play a significant role when dealing with angles and their ratios. They are the reverse operations of the standard trigonometric functions like sine, cosine, and tangent.
Instead of finding the ratio from an angle, they help find the angle from a given ratio. For example,

• \( \sin^{-1} \) or arcsin is used when you have the sine ratio and need the angle.
• \( \cos^{-1} \) or arccos is used when you have the cosine ratio and need the angle.

In the given exercise,

• \( \sin^{-1} \frac{3}{5} \) determines the angle whose sine is 3/5. The result can be denoted as \( \theta \).
• \( \cos^{-1} \frac{5}{13} \) determines the angle whose cosine is 5/13. The result can be denoted as \( \beta \).

Once identified, these angles can be used with other trigonometric identities to find the required solution, as seen in the process of determining \( \cos(\theta + \beta) \).

Exact Values

Exact values in trigonometry allow us to resolve expressions without relying on decimal approximations. When finding the precise value of trigonometric expressions, we often revert to fundamental properties and identities.
In the problem at hand, determining exact trigonometric values involves finding the complementary or supplementary trigonometric values based on known ratios.

• Given \( \sin \theta = \frac{3}{5} \), we find \( \cos \theta \) using the Pythagorean identity: \( \cos^2 \theta = 1 - \sin^2 \theta \), leading to \( \cos \theta = \frac{4}{5} \).
• Similarly, \( \cos \beta = \frac{5}{13} \) was used to find \( \sin \beta \) in the same manner, resulting in \( \sin \beta = \frac{12}{13} \).

These exact values are crucial for accurately calculating the resultant trigonometric expression \( \cos(\theta + \beta) \) through the cosine sum formula.

Cosine Sum Formula

The cosine sum formula is essential in trigonometry for combining angles. It expresses the cosine of the sum of two angles in terms of their individual sines and cosines.
The formula is:

• \( \cos(\theta + \beta) = \cos \theta \cos \beta - \sin \theta \sin \beta \).

In the provided exercise, this formula was critical in resolving the final expression:

• Using known values: \( \cos \theta = \frac{4}{5} \) and \( \cos \beta = \frac{5}{13} \), and the previously calculated \( \sin \theta = \frac{3}{5} \) and \( \sin \beta = \frac{12}{13} \).
• The substitution into the formula gives \( \frac{4}{5} \times \frac{5}{13} - \frac{3}{5} \times \frac{12}{13} \).
• Solving these multiplications and performing a straightforward subtraction leads us to the exact answer: \( \frac{-16}{65} \).

Understanding and applying this formula is key to solving expressions that involve angles added together without the use of a calculator.