Problem 49 - Algebra and Trigonometry

Question

\(47-50\) Find the exact value of the given expression. $$ \sec \left(2 \sin ^{-1} \frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{8}{7} \).

Step 1: Understand the Problem

We need to find the exact value of \( \sec \left(2 \sin ^{-1} \frac{1}{4}\right) \). This involves understanding how to apply trigonometric identities to compute the expression.

Step 2: Express the Problem in Terms of Known Angles

Let \( \theta = \sin^{-1} \frac{1}{4} \). This means \( \sin \theta = \frac{1}{4} \). We are required to find \( \sec(2\theta) \).

Step 3: Use a Trigonometric Identity

We know that \( \sec A = \frac{1}{\cos A} \). We also know that \( \cos(2\theta) = 1 - 2\sin^2 \theta \) or \( \cos(2\theta) = 2\cos^2 \theta - 1 \). We will use the identity \( \cos(2\theta) = 1 - 2 \sin^2 \theta \).

Step 4: Calculate \( \sin^2 \theta \)

Since \( \sin \theta = \frac{1}{4} \), \( \sin^2 \theta = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \).

Step 5: Substitute \( \sin^2 \theta \) into Identity

Now substitute \( \sin^2 \theta = \frac{1}{16} \) into \( \cos(2\theta) = 1 - 2\sin^2 \theta \):\[ \cos(2\theta) = 1 - 2 \times \frac{1}{16} = 1 - \frac{2}{16} = 1 - \frac{1}{8} = \frac{7}{8} \].

Step 6: Find \( \sec(2\theta) \)

Since \( \sec(2\theta) = \frac{1}{\cos(2\theta)} \), substitute \( \cos(2\theta) = \frac{7}{8} \):\[ \sec(2\theta) = \frac{1}{\frac{7}{8}} = \frac{8}{7} \].

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in finding angles when the value of a trigonometric ratio is known. In the context of our exercise, we have the inverse sine function, represented as \( \sin^{-1} \).

• \( \sin^{-1} \left( \frac{1}{4} \right) \) gives the angle whose sine is \( \frac{1}{4} \).
• When you denote \( \theta = \sin^{-1} \left( \frac{1}{4} \right) \), it simply translates to saying that \( \sin \theta = \frac{1}{4} \).

Using inverse trigonometric functions is fundamental when working with angle identities as it forms the basis to calculate other values based on known equations. This angle, \( \theta \), will then be used to discover other trigonometric ratios and expressions.

Secant Function

The secant function is one of the six fundamental trigonometric functions, represented as \( \sec \). It is defined as the reciprocal of the cosine function:

• \( \sec A = \frac{1}{\cos A} \)

In our solution, we are interested in \( \sec(2\theta) \), where \( \theta \) is an angle derived from the inverse sine function. Thus, once \( \cos(2\theta) \) is found, the next step is calculating \( \sec(2\theta) \) as follows:

• We determine \( \cos(2\theta) \), and then \( \sec(2\theta) = \frac{1}{\cos(2\theta)} \).

This function is particularly useful in problems involving identities and transformations, where ratios of adjacent and hypotenuse arise, thus helping us evaluate more complex expressions.

Double Angle Formulas

Double angle formulas are trigonometric identities used to express trigonometric functions of \( 2\theta \) in terms of \( \theta \). There are specific formulas for sine, cosine, and tangent double angles. In this exercise, we use the cosine double angle formula:

• \( \cos(2\theta) = 1 - 2 \sin^2 \theta \)
• Alternatively, \( \cos(2\theta) = 2 \cos^2 \theta - 1 \)

For this problem, we're using \( \cos(2\theta) = 1 - 2 \sin^2 \theta \), which helps us find \( \cos \) of double the angle given by the inverse function. After substituting \( \sin^2 \theta = \frac{1}{16} \), we find:

• \( \cos(2\theta) = 1 - \frac{1}{8} = \frac{7}{8} \).

These formulas are critical in simplifying expressions and solving equations as they provide a direct way to compute the compatible functional value, utilizing known values effectively.