Problem 49 - Precalculus Mathematics for Calculus

Question

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: Understanding the Component Parts

The expression we're dealing with is \( \sec \left(2 \sin^{-1} \frac{1}{4}\right) \). Here, we need to find the angle \( \theta \) such that \( \sin \theta = \frac{1}{4} \). This \( \theta \) is actually the input for \( \sin^{-1} \), and it lies within the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Step 2: Using Double Angle Identities

Recall the double angle formula for cosine since secant is the reciprocal of cosine:\[\cos(2\alpha) = 1 - 2 \sin^2(\alpha).\]We have \( \alpha = \sin^{-1} \frac{1}{4} \). Thus, \( \sin \alpha = \frac{1}{4} \), so substituting gives us:\[\cos(2\alpha) = 1 - 2\left(\frac{1}{4}\right)^2.\]

Step 3: Simplifying the Function

Continue simplifying \( \cos(2\alpha) \):\[\cos(2\alpha) = 1 - 2 \frac{1}{16} = 1 - \frac{1}{8} = \frac{7}{8}.\]

Step 4: Calculating Secant

Secant is the reciprocal of the cosine function. Therefore, \[\sec(2\alpha) = \frac{1}{\cos(2\alpha)} = \frac{1}{\frac{7}{8}} = \frac{8}{7}.\]

Step 5: Conclusion

Thus, the exact value of the expression \( \sec \left(2 \sin^{-1} \frac{1}{4}\right) \) is \( \frac{8}{7} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential when we want to determine the angle associated with a particular trigonometric value. These functions help us "invert" the typical trigonometric relationships. For example, the inverse sine function, written as \( \sin^{-1}(x) \), returns an angle \( \theta \) for which \( \sin(\theta) = x \). In our exercise, we need to find the angle \( \theta \) such that \( \sin \theta = \frac{1}{4} \).
This means solving \( \sin^{-1}(\frac{1}{4}) \), which gives us the value of \( \alpha \) (or \( \theta \)) used later. It's key to remember that the output of \( \sin^{-1} \) is always within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This is crucial as it ensures that we get the correct angle position when applying further identities.

• \( \sin^{-1}(x): \text{Finds the angle for which sine of that angle equals } x \)
• \( \text{Output Range: } [-\frac{\pi}{2}, \frac{\pi}{2}] \)

Double Angle Formulas

The double angle formulas are a set of trigonometric identities that express trigonometric functions of double angles \(2\alpha\) in terms of single angle \(\alpha\). These formulas are particularly useful when simplifying expressions or solving equations involving trigonometric functions.
One key formula that is used in our solution is the double angle formula for cosine: \[\cos(2\alpha) = 1 - 2\sin^2(\alpha).\]With \( \alpha = \sin^{-1}\left(\frac{1}{4}\right) \), we have \( \sin(\alpha) = \frac{1}{4} \). Substituting into the formula gives us:\[\cos(2\alpha) = 1 - 2\left(\frac{1}{4}\right)^2.\] This formula helps reduce our expression to a more manageable form and is a standard identity used in many trigonometric problems.

• \( \cos(2\alpha) = 1 - 2\sin^2(\alpha) \)
• \( \text{Used for expressions like } \sec(2\alpha) \text{ by finding the corresponding cosine value} \)

Secant Function

The secant function, denoted as \( \sec(\theta) \), is one of the six primary trigonometric functions and is defined as the reciprocal of the cosine function: \[\sec(\theta) = \frac{1}{\cos(\theta)}.\] In trigonometric problems, secant can often be introduced through such reciprocal relationships, especially when given expressions like \( \sec(2\alpha) \).
The exercise expression \( \sec \left(2 \sin^{-1} \frac{1}{4} \right) \) was simplified to find \( \cos(2\alpha) = \frac{7}{8} \). Therefore, finding \( \sec(2\alpha) \) involves taking the reciprocal, yielding:\[\sec(2\alpha) = \frac{1}{\frac{7}{8}} = \frac{8}{7}.\] This approach is central whenever secant is involved, as understanding reciprocals is key.

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \text{Reciprocal relationships simplify the problem-solving process}\)

Cosine Function

The cosine function, \( \cos(\theta) \), is another fundamental trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It plays a pivotal role in many trigonometric expressions and identities, like the double angle formulas.
In our exercise, cosine is crucial, as secant is its reciprocal. We focus on \( \cos(2\alpha) \) with the double angle formula: \[\cos(2\alpha) = 1 - 2\sin^2(\alpha).\] The double angle formula provided us with a straightforward path to simplify and understand the situation by reducing the expression to find \( \cos(2\alpha) = \frac{7}{8} \).
Understanding this process is important for not only solving similar problems but also grasping the interdependencies between trigonometric functions.

• \( \cos(\theta): \text{Ratio of adjacent/hypotenuse} \)
• \( \cos(2\alpha) = 1 - 2\sin^2(\alpha); \text{Key for simplifying secant expressions} \)