Problem 22 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sec \left(\tan ^{-1} 4\right) $$

Step-by-Step Solution

Answer

The exact value is \( \sqrt{17} \).

Step 1: Understand the Inverse Trigonometric Function

The expression involves \( \tan^{-1} 4 \). This means we are dealing with an angle \( \theta \) such that \( \tan \theta = 4 \).

Step 2: Draw a Right Triangle

Imagine a right triangle where \( \tan \theta = \frac{opposite}{adjacent} = 4 \). We can choose the opposite side to be 4 and the adjacent side to be 1. This choice is arbitrary but simplifies our calculations.

Step 3: Find the Hypotenuse Using the Pythagorean Theorem

The hypotenuse \( h \) can be found using the Pythagorean theorem: \( h = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \).

Step 4: Relate Secant to the Triangle Sides

Recalling that \( \sec \theta \) is the reciprocal of \( \cos \theta \), and \( \cos \theta = \frac{adjacent}{hypotenuse} \), we find \( \sec \theta = \frac{1}{\cos \theta} = \frac{h}{adjacent} = \frac{\sqrt{17}}{1} = \sqrt{17} \).

Step 5: Conclusion

Thus, the expression \( \sec \left(\tan^{-1} 4\right) = \sqrt{17} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry. They express relationships between the different trigonometric functions. These identities make it simpler to solve trigonometric equations and simplify expressions.

Among the basic identities, we often use the reciprocal identities. For example:

• \( ext{sec } \theta = \frac{1}{\cos \theta} \)
• \( ext{cosec } \theta = \frac{1}{\sin \theta} \)
• \( ext{cot } \theta = \frac{1}{\tan \theta} \)

These identities help convert complex trigonometric expressions into simpler ones. Moreover, using compound angle identities or double angle formulas can help solve more challenging problems.

In our example, we used the fact that \( \sec \theta \) is the reciprocal of \( \cos \theta \), which helped us find the value of the trigonometric expression without needing a calculator. Remember, understanding these identities is crucial for tackling inverse trigonometric functions.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of all geometric principles involving right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The formula is expressed as: \[ a^2 + b^2 = c^2\]where:

• \(a\) and \(b\) are the lengths of the triangle's two legs
• \(c\) is the length of the hypotenuse

In our problem, we had a right triangle where the sides were chosen based on the tangent function: the opposite was 4, and the adjacent was 1. By applying the Pythagorean Theorem, we calculated the hypotenuse as \( \sqrt{17} \). This formula is essential for deriving trigonometric ratios of right triangles, especially when parameters like \( \tan \theta \) are given directly.

Right Triangle

Understanding the structure of a right triangle is essential for solving trigonometric problems. In a right triangle, one of the angles is always 90 degrees and the other two angles are acute.

Here’s why right triangles are so crucial:

• Their geometric properties allow the use of the Pythagorean Theorem.
• Each of the trigonometric functions—sine, cosine, tangent, etc.—can be represented as ratios of the right triangle's sides.

In the given problem, we assigned values to the triangle's sides such that the ratio \( \tan \theta = \frac{4}{1} \). We then used this conceptual triangle to derive \( \sec \theta \), which is \( \frac{hypotenuse}{adjacent} \). The simplicity and clarity of right triangles enable us to break down complex trigonometric problems into understandable steps.