Problem 90 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The exact real number value is \( \frac{16}{65} \).

Step 1: Understand what is given

We are given the expression \( \cos \left(\cos^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13}\right) \). This involves finding the exact value of the cosine of a sum of angles defined using inverse trigonometric functions.

Step 2: Identify angles using inverse functions

The expression \( \cos^{-1} \frac{3}{5} \) represents an angle \( \theta \) in a right triangle where \( \cos \theta = \frac{3}{5} \). Similarly, \( \sin^{-1} \frac{5}{13} \) represents an angle \( \phi \) where \( \sin \phi = \frac{5}{13} \).

Step 3: Find remaining triangle sides

For \( \theta \), we have \( \cos \theta = \frac{3}{5} \) thus the opposite side is \( \sqrt{5^2 - 3^2} = \sqrt{16} = 4 \). For \( \phi \), we have \( \sin \phi = \frac{5}{13} \) thus the adjacent side is \( \sqrt{13^2 - 5^2} = \sqrt{144} = 12 \).

Step 4: Compute sine and cosine using triangle sides

Now, \( \sin \theta = \frac{4}{5} \) and \( \cos \phi = \frac{12}{13} \) using the sides calculated in the previous step.

Step 5: Use angle sum identity for cosine

The cosine of a sum of angles is given by the identity: \( \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \). Substitute the values: \( \cos \theta = \frac{3}{5}, \cos \phi = \frac{12}{13}, \sin \theta = \frac{4}{5}, \sin \phi = \frac{5}{13} \).

Step 6: Substitute values into the identity

Plug the values into the identity: \[ \cos(\theta + \phi) = \left(\frac{3}{5}\right)\left(\frac{12}{13}\right) - \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) \].

Step 7: Simplify the expression

Compute the expression: \[ \cos(\theta + \phi) = \frac{36}{65} - \frac{20}{65} = \frac{16}{65} \].

Step 8: Result interpretation

Thus, the exact real number value of the expression is \( \frac{16}{65} \).

Key Concepts

Cosine Identity

The cosine identity is a fundamental element of trigonometry, especially handy when working with angle sums and differences. The identity for the cosine of the sum of two angles is particularly useful: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]This formula helps determine the cosine of an angle composed of two smaller, simpler angles. When solving trigonometric problems involving inverse functions, this identity is incredibly beneficial.
Inverse trigonometric problems often break down complex angles into familiar components, and employing the cosine identity helps reintegrate these components. In our given exercise, once angles \( \theta \) and \( \phi \) were determined using the inverse functions, the cosine identity was used to derive a comprehensive solution."
Understanding how identities like these work is key to solving similar trigonometric expressions efficiently.

Angle Sum Formula

Trigonometric functions have specific formulas for computing the sine, cosine, and tangent of the sum of two angles. For cosine, the angle sum formula is:\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]This formula tells us how to find the cosine of a combined angle \( a + b \) using the trig values of the separate angles, \( a \) and \( b \).
In our exercise, the angles were found using inverse trigonometric functions. Here, \( \cos^{-1} \frac{3}{5} \) was converted to \( \theta \) with cosine \( \theta = \frac{3}{5} \), and \( \sin^{-1} \frac{5}{13} \) to \( \phi \) with sine \( \phi = \frac{5}{13} \).
This approach simplifies the expression considerably, allowing us to apply the formula to calculate \( \cos(\theta + \phi) \), resulting in an exact value without a calculator.

Right Triangle Trigonometry

Right triangle trigonometry is foundational for understanding how trigonometric functions like sine, cosine, and tangent relate to angle measures within right triangles.
When we use inverse trigonometric functions such as \( \cos^{-1} \) or \( \sin^{-1} \), it involves identifying and interpreting angles within a right triangle based on given ratios for the sides. For instance, with \( \cos \theta = \frac{3}{5} \), this implies a right triangle with adjacent \( 3 \), hypotenuse \( 5 \), and after computation, an opposite side of \( 4 \).
Similarly, for \( \sin \phi = \frac{5}{13} \), creating a triangle based on the given sine value results in an opposite \( 5 \) and hypotenuse \( 13 \), leading to the adjacent side equating to \( 12 \).
Right triangle trigonometry thus plays a crucial role in applying inverse trigonometric functions, allowing us to dissect and reconstruct angle relations to solve complex trigonometric problems.