Problem 101 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$\cos \left(\tan ^{-1} \frac{4}{3}+\cos ^{-1} \frac{5}{13}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \cos \left(\tan^{-1} \frac{4}{3} + \cos^{-1} \frac{5}{13} \right) \) is \( -\frac{33}{65} \).

Step 1: Convert Inverse Tangent to an Angle

Let's start with \( \tan^{-1} \frac{4}{3} \). We can interpret this as the angle whose tangent is \( \frac{4}{3} \). Consider a right triangle where the opposite side is 4 and the adjacent side is 3. By the Pythagorean theorem, the hypotenuse would be \( \sqrt{4^2 + 3^2} = 5 \).

Step 2: Convert Inverse Cosine to an Angle

Now let's consider \( \cos^{-1} \frac{5}{13} \). This represents the angle whose cosine is \( \frac{5}{13} \). Consider a right triangle where the hypotenuse is 13 and the adjacent side is 5. By the Pythagorean theorem, the opposite side would be \( \sqrt{13^2 -5^2} = 12 \).

Step 3: Find the Cosine of the Sum

Now we can rewrite the original expression as \( \cos \left(\tan^{-1} \frac{4}{3} + \cos^{-1} \frac{5}{13} \right) \) = \( \cos \left(\angle A + \angle B \right) \), where A is the angle from Step 1 and B is the angle from Step 2. We use the identity for the cosine of the sum of two angles, which is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). Using the sides of the triangles from Steps 1 and 2, we substitute the respective cos and sin values into the formula. This gives \( \left(\frac{3}{5} \times \frac{5}{13} \right) - \left(\frac{4}{5} \times \frac{12}{13} \right) \). Simplifying this we get \( \frac{15}{65} - \frac{48}{65} = -\frac{33}{65} \).