Problem 48 - Precalculus Mathematics for Calculus

Question

Find the exact value of the given expression. $$\cos \left(2 \tan ^{-1} \frac{12}{3}\right)$$

Step-by-Step Solution

Answer

The value is \( -\frac{15}{17} \).

Step 1: Simplify the Inverse Tangent Expression

First, simplify the inverse tangent expression \( \tan^{-1} \left( \frac{12}{3} \right) \). This simplifies to \( \tan^{-1}(4) \) because \( \frac{12}{3} = 4 \).

Step 2: Consider Trigonometric Identity for Double Angle Formula

Use the double angle formula for cosine: \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). Here \( \theta = \tan^{-1}(4) \).

Step 3: Represent \( \theta \) in a Triangle

Since \( \theta = \tan^{-1}(4) \), let \( \tan(\theta) = 4 \). This means in a right triangle, the opposite side is 4 and the adjacent side is 1. Hence, the hypotenuse is \( \sqrt{4^2 + 1^2} = \sqrt{17} \).

Step 4: Calculate \( \sin(\theta) \) and Substitute

From the triangle, we have \( \sin(\theta) = \frac{4}{\sqrt{17}} \). Now use this in the formula: \( \cos(2\theta) = 1 - 2\left(\frac{4}{\sqrt{17}}\right)^2 = 1 - 2\left(\frac{16}{17}\right) \).

Step 5: Simplify the Expression

Simplify the expression \( 1 - 2 \times \frac{16}{17} = 1 - \frac{32}{17} = \frac{17 - 32}{17} = \frac{-15}{17} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find an angle when given a trigonometric ratio. In our exercise, we came across the function \( \tan^{-1}(4) \), which is the inverse of the tangent function and finds the angle whose tangent is 4.

• Understanding Inverse Functions: Inverse functions reverse the effect of the original function. For example, if \( \tan(\theta) = 4 \), then \( \tan^{-1}(4) = \theta \).


• Application in Right Triangles: Considering \( \tan(\theta) = 4 \), we use a right triangle to visualize this. The opposite side over the adjacent side is 4:1, meaning the opposite side is 4 units long and the adjacent is 1 unit. The hypotenuse is calculated using the Pythagorean theorem: \( \sqrt{4^2 + 1^2} = \sqrt{17} \).

Inverse trigonometric functions are pivotal in calculating angles, which we further use in double angle formulas to solve expressions like \( \cos \left( 2 \tan^{-1} 4 \right) \).

Double Angle Formulas

Double angle formulas are a set of trigonometric identities that are useful for expressing trigonometric functions of double angles (i.e., \( 2\theta \)), in terms of functions of single angles.

• Cosine Double Angle Formula: For cosine, it is expressed as \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). This formula is handy because it allows us to work with angles of \( 2\theta \) by knowing \( \sin(\theta) \).


• Why Use Double Angle Formulas: In our problem, the goal was to find \( \cos \left( 2 \tan^{-1} 4 \right) \). By converting the angle \( \tan^{-1} 4 \) into known trigonometric values and substituting into this formula, it allows us to solve the expression.

This process simplifies our calculation by leveraging known identities, making it much simpler than directly finding angles of more than 90 degrees without a calculator.

Trigonometric Expressions

Trigonometric expressions involve the use of trigonometric functions to represent angles and sides of triangles. They often require simplification using identities to find exact values, like those seen with inverse functions and double angles.

• Breaking Down Expressions: In our example, the trigonometric expression involved \( \cos(2\theta) \), where \( \theta \) had to be found through an inverse function. Step by step, each piece was simplified until the entire expression was solvable.


• Importance in Problem Solving: Trigonometric expressions often appear in different areas of mathematics and physics, requiring a solid understanding of identities. Our problem illustrated this, as complex expressions were broken down into simpler parts to find the result \( \frac{-15}{17} \).

Thus, mastering the simplification and transformation of trigonometric expressions is essential in tackling more advanced mathematical problems.