Problem 57 - Precalculus: A Concise Course

Question

Find the exact value of the expression. (Hint: Sketch a right triangle.) $$ \cos \left(\tan ^{-1} 2\right) $$

Step-by-Step Solution

Answer

\(\cos \left(\tan ^{-1} 2\right) = \frac{1}{\sqrt{5}}\)

Step 1: Define the Inverse Trigonometric Function

We start by considering \(\tan^{-1}(2)\). This is the angle that has a tangent of \(2\). By definition, the tangent of an angle is the ratio of the side opposite to the angle to the adjacent side. So create a right triangle such that the side opposite the angle is \(2\) (opposite side) and the adjacent side is \(1\) (adjacent side).

Step 2: Calculate the Hypotenuse

Since we have a right triangle, we can use the Pythagorean Theorem to calculate the length of the hypotenuse. According to the Pythagorean theorem, \(\text{Hypotenuse} = \sqrt{(\text{Opposite side})^2 + (\text{Adjacent side})^2} = \sqrt{2^2 + 1^2} = \sqrt{5}\).

Step 3: Calculate the Cosine

Cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. So, \(\cos (\tan^{-1}2) = \frac{1}{\sqrt{5}}\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when given specific trigonometric values. The function \( \tan^{-1}(x) \) finds the angle whose tangent is \( x \). In our exercise, \( \tan^{-1}(2) \) gives us an angle with a tangent of \( 2 \). This means the ratio of the opposite side to the adjacent side of this angle is \( 2:1 \).

These functions are essential when working backward from the trigonometric ratio to the angle. They are widely used in geometry, physics, and engineering.

• Tangent: Opposite/Adjacent
• Using Inverse: Helps find the angle
• Application: Ideal for solving real-world problems

Understanding this concept allows us to build and solve triangles based on ratios rather than angles or side lengths directly.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone in geometry, especially with right triangles. It states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the hypotenuse (the side opposite the right angle).

In formula terms, it's expressed as \( a^2 + b^2 = c^2 \).

• Determine Missing Side: Useful in calculating an unknown side length
• Right Triangles Only: Applicable solely to right triangles
• Real-World Uses: Engineering, architecture, and physics

In our exercise, when the opposite side is \( 2 \) and the adjacent is \( 1 \), the hypotenuse can be calculated as \( \sqrt{5} \), establishing the triangle's necessary dimensions to solve for the cosine value.

Right Triangle

A right triangle is defined by having one 90-degree angle. Due to this right angle, the relationships between the sides are uniquely established, allowing the use of trigonometric ratios like sine, cosine, and tangent.

Key characteristics of right triangles include:

• Hypotenuse: Always the longest side, opposite the right angle
• Legs: The two shorter sides adjacent to the right angle
• Trigonometric Ratios: Relate angles to side lengths

When sketching a right triangle to solve for \( \cos(\tan^{-1}2) \), one can visualize the triangle with sides \( 2 \) and \( 1 \). This setup aids in calculating trigonometric values effectively, leveraging the relationships defined by the triangle's geometry.