Problem 29 - Precalculus Mathematics for Calculus

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: Understand the Problem

The exercise asks for the value of \(\sec \left(\sin^{-1} \frac{12}{13}\right)\). This expression requires us to find the secant of an angle whose sine is \(\frac{12}{13}\).

Step 2: Find the Angle

Let \( \theta = \sin^{-1} \frac{12}{13} \). This means \( \sin \theta = \frac{12}{13} \). \(\theta\) is an angle in a right triangle where the opposite side is 12 and the hypotenuse is 13.

Step 3: Use Pythagorean Theorem to Find the Adjacent Side

Use the Pythagorean theorem to find the adjacent side of the triangle: \[ x = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5 \]The adjacent side to \(\theta\) is 5.

Step 4: Find Secant of the Angle

\(\sec \theta\) is the reciprocal of \(\cos \theta\). So, \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13} \] Hence, \[ \sec \theta = \frac{1}{\cos \theta} = \frac{13}{5} \].

Step 5: Conclude with the Exact Value

Therefore, the exact value of the expression \(\sec \left(\sin^{-1} \frac{12}{13}\right)\) is \(\frac{13}{5}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the measure of an angle when given the value of a trigonometric ratio. Functions such as \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \) are the inverses of the standard sine, cosine, and tangent functions, respectively.

Let's consider the example \( \sin^{-1} \left(\frac{12}{13}\right)\). Here, we're looking for an angle \( \theta \) such that its sine is \( \frac{12}{13} \). This means we're trying to find \( \theta \) that makes \( \sin \theta = \frac{12}{13} \).

In a right triangle, sine is defined as the ratio of the length of the opposite side to the hypotenuse. The inverse function helps us find which angle results in this specific ratio. It's an important concept in trigonometry, often used to solve problems related to finding angles when sides of the triangle are known.

Secant Function

The secant function, denoted as \( \sec \), is one of the six basic trigonometric functions. It is closely related to the cosine function.

Specifically, the secant of an angle \( \theta \) is the reciprocal of the cosine of that angle. Mathematically, it is expressed as:

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

This means that to find \( \sec \theta \), you first need to find \( \cos \theta \). For an angle \( \theta \) in a right triangle, \( \cos \theta \) is the ratio of the length of the adjacent side to the hypotenuse.

Applying this to the exercise, after determining the adjacent side using the Pythagorean theorem, we calculated \( \cos \theta \) as \( \frac{5}{13} \). Thus, \( \sec \theta = \frac{1}{ \cos \theta } = \frac{13}{5} \). That's the value we were seeking, using the property of the secant function.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in trigonometry and geometry. It relates the sides of a right triangle.

The theorem states that in a right triangle, the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \). Expressed as:

• \[ c^2 = a^2 + b^2 \]

In our example, we knew the hypotenuse and one side (opposite) of a right triangle. The sine was given as \( \frac{12}{13} \), meaning the opposite side was 12, and the hypotenuse was 13.

We used the theorem to find the missing side, the adjacent one. Plugging into the equation, we had:

\[ 13^2 = 12^2 + x^2 \]
Solving for \( x \), we got \( x = 5 \). This important step allowed us to compute \( \cos \theta \) and ultimately find \( \sec \theta \). This makes the Pythagorean theorem a powerful tool when analyzing right triangles and trigonometric values.