Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(2 \sin ^{-1} \frac{1}{4}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Inverse Trigonometric Function

First, we identify what \( \sin^{-1} \frac{1}{4} \) means. It is an angle \( \theta \) such that \( \sin \theta = \frac{1}{4} \). This angle \( \theta \) lies within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Use Trigonometric Identity to Find Cosine of Double Angle

We want to find \( \cos \left(2 \sin^{-1} \frac{1}{4}\right) \). Utilize the double angle identity for cosine: \( \cos(2\alpha) = 1 - 2\sin^2(\alpha) \). Here, \( \alpha = \sin^{-1} \frac{1}{4} \).

Step 3: Find \(\sin^2(\alpha)\)

We know \(\sin(\alpha) = \frac{1}{4} \). Therefore, \(\sin^2(\alpha) = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \).

Step 4: Apply the Double Angle Identity

Substitute \(\sin^2(\alpha) = \frac{1}{16} \) into the identity: \( \cos(2\alpha) = 1 - 2 \times \frac{1}{16} \). Simplify this to get \( \cos(2\alpha) = 1 - \frac{2}{16} = 1 - \frac{1}{8} = \frac{7}{8} \).

Step 5: Conclusion

After simplifying, the exact real value of the given trigonometric expression is \( \frac{7}{8} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry that help us find angles when given sine, cosine, or tangent values. The inverse sine, written as \( \sin^{-1} \), takes a real number input and returns an angle. This angle is specifically chosen from the principal value range. For \( \sin^{-1} \), the angle is always within

• \([-\frac{\pi}{2}, \frac{\pi}{2} ]\) for radians,
• or [-90^\circ, 90^\circ] for degrees.

For example, \( \sin^{-1} \frac{1}{4} \) means finding an angle \( \theta \) such that \( \sin \theta = \frac{1}{4} \). Understanding this process is crucial when using inverse trigonometric functions in more complex trigonometric expressions, like double angles.

Trigonometric Identities

Trigonometric identities are fundamental equations that relate the trigonometric functions to one another. They are vital when simplifying expressions or solving equations involving trigonometric functions. These identities include:

• Pythagorean identities: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle sum and difference identities.
• Double and half-angle formulas.

When we solve \( \cos(2 \sin^{-1} \frac{1}{4}) \), we apply the double angle identity for cosine. The identity \( \cos(2\alpha) = 1 - 2\sin^2(\alpha) \) allows us to express \( \cos \) of a double angle in terms of \( \sin \) of a single angle. This method is beneficial since converting all functions to one base value simplifies calculations and ensures accuracy.

Double Angle Formulas

Double angle formulas involve expressions which double the angle of the trigonometric function. Double angles simplify the function's calculations and are especially handy in solving trigonometric problems without a calculator. For cosine, the double angle formula is:

• \( \cos(2\alpha) = 1 - 2\sin^2(\alpha) \)
• Another version: \( \cos(2\alpha) = 2\cos^2(\alpha) - 1 \)

These formulas transform complex expressions into simpler ones. In the exercise, calculating \( \cos(2 \sin^{-1} \frac{1}{4}) \) requires knowing \( \sin^2(\alpha) = \frac{1}{16} \), simplifying it further using the identity: \[\cos(2\alpha) = 1 - 2 \left(\frac{1}{16}\right) = \frac{7}{8}\]This demonstrates the power and versatility of double angle formulas in trigonometry.