Problem 29 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \sec \left(\sin ^{-1} \frac{12}{13}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{13}{5} \).

Step 1: Understanding the Inverse Sine Function

The expression involves \( \sin^{-1} \left( \frac{12}{13} \right) \), which is the angle whose sine is \( \frac{12}{13} \). Let's call this angle \( \theta \). Therefore, \( \sin \theta = \frac{12}{13} \).

Step 2: Using the Pythagorean Identity

We use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{12}{13} \):\[\left(\frac{12}{13}\right)^2 + \cos^2 \theta = 1\]Solve for \( \cos^2 \theta \):\[\frac{144}{169} + \cos^2 \theta = 1\]\[\cos^2 \theta = 1 - \frac{144}{169} = \frac{25}{169}\]Take the square root to find \( \cos \theta \):\( \cos \theta = \pm \frac{5}{13} \).

Step 3: Determine the Correct Sign of Cosine

Since \( \theta = \sin^{-1} \left( \frac{12}{13} \right) \), it is in the first quadrant where both sine and cosine are positive. So, \( \cos \theta = \frac{5}{13} \).

Step 4: Calculate the Secant

Secant is the reciprocal of cosine. Thus:\[\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}\]Therefore, \( \sec \left( \sin^{-1} \frac{12}{13} \right) = \frac{13}{5} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the operations of the basic trigonometric functions like sine, cosine, tangent, etc. They are essential in trigonometry because they allow us to find the angle when we know the trigonometric value. For instance, in our exercise, we have the inverse sine function represented as \( \sin^{-1} \left( \frac{12}{13} \right) \). This expression asks us to find the angle \( \theta \) such that the sine of \( \theta \) is \( \frac{12}{13} \).

When dealing with inverse trigonometric functions, it is important to consider their range and domain. The output, or range, of the inverse sine, \( \sin^{-1} \), falls between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which corresponds to angles in the first and fourth quadrants. Knowing this helps determine the signs of other related trigonometric functions.

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry, expressing the unity of sine and cosine squares. It is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is invaluable for solving trigonometric problems, such as in our exercise, where \( \sin \theta \) is \( \frac{12}{13} \). To find \( \cos \theta \), we substitute into the identity:

• The square of \( \sin \theta \), \( \left( \frac{12}{13} \right)^2 = \frac{144}{169} \).
• Thus, \( \cos^2 \theta = 1 - \frac{144}{169} = \frac{25}{169} \).
• Taking the square root gives \( \cos \theta = \pm \frac{5}{13} \).

When determining the sign of the cosine based on quadrant information, knowing the range of the inverse function allows us to choose the positive value since the angle is in the first quadrant.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived by taking the reciprocal of the basic trigonometric functions. The secant function, for example, is the reciprocal of the cosine function. In formulas, this is shown as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In the exercise, after finding \( \cos \theta \) as \( \frac{5}{13} \), calculating \( \sec \theta \) requires taking its reciprocal:
\( \sec \theta = \frac{1}{\frac{5}{13}} = \frac{13}{5} \).

Understanding these reciprocal relationships is key in trigonometry as they connect different functions and allow for flexible problem-solving approaches. They are often used when switching between angles or working in various math areas like calculus or geometry.