Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(2 \cos ^{-1} \frac{1}{5}\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{4\sqrt{6}}{25} \).

Step 1: Understand the Given Expression

The question involves finding the value of \( \sin \left(2 \cos^{-1} \frac{1}{5} \right) \). The \( \cos^{-1} \) function is the inverse cosine, which will yield an angle whose cosine is \( \frac{1}{5} \). The expression then requires finding the sine of twice this angle.

Step 2: Define the Inverse Cosine Output

Let \( \theta = \cos^{-1} \frac{1}{5} \). This means that \( \cos \theta = \frac{1}{5} \). We need to determine additional trigonometric values based on this relationship.

Step 3: Use Pythagorean Identity to Find Sine of Theta

From the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), solve for \( \sin \theta \): \( \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left( \frac{1}{5} \right)^2 = 1 - \frac{1}{25} = \frac{24}{25} \). Thus, \( \sin \theta = \sqrt{\frac{24}{25}} = \frac{\sqrt{24}}{5} = \frac{2\sqrt{6}}{5} \), considering \( \sin \theta \) is positive in the principal range of \( \cos^{-1} \) (between 0 and \( \pi \)).

Step 4: Apply Angle Doubling Formula

Apply the double angle formula for sine: \( \sin(2\theta) = 2\sin \theta \cos \theta \). Substitute the known values: \( \sin(2\theta) = 2 \times \frac{2\sqrt{6}}{5} \times \frac{1}{5} = \frac{4\sqrt{6}}{25} \).

Step 5: Simplify the Result

The calculated value of the expression is \( \sin(2\theta) = \frac{4\sqrt{6}}{25} \). This is the exact real number value of the given trigonometric expression.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse processes of the regular trigonometric functions. They are used to find an angle when we know the value of a trigonometric ratio. For instance, if we know that

• \( \cos \theta = \frac{1}{5} \),

then the inverse cosine function, denoted as \( \cos^{-1}(x) \), can find the angle theta. This is commonly referred to as the arc cosine. The range of the inverse cosine function is from 0 to \( \pi \). This means that it always provides angles in that range, which are the principal values for cosine.
If \( \theta = \cos^{-1}(\frac{1}{5}) \), \( \theta \) is an angle such that the cosine of \( \theta \) is \( \frac{1}{5} \). Remember, using inverse trigonometric functions is pivotal whenever you have to work backward from a trigonometric value to find the related angle.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry, expressed as:

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

This helps us find the sine of an angle if we know the cosine, and vice versa. It's derived from the Pythagorean theorem applied to a unit circle, where the sum of the squares of the legs of the triangle formed (sine and cosine of the angle) equals the square of the hypotenuse (which is 1).
In our example, where \( \cos \theta = \frac{1}{5} \), we use this identity to find \( \sin \theta \) by rearranging it to:

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

After substituting \( \frac{1}{5} \) for \( \cos \theta \), we calculate \( \sin^2 \theta = \frac{24}{25} \) and thus \( \sin \theta = \frac{2\sqrt{6}}{5} \). This identity is very useful when combining or converting between trigonometric functions.

Double Angle Formulas

Double angle formulas allow us to express trigonometric functions of double angles (like \( 2\theta \)) in terms of functions of single angles (\( \theta \)). These formulas are essential for simplifying expressions where an angle is twice another.
The double angle formula for sine is given by:

• \( \sin(2\theta) = 2\sin \theta \cos \theta \).

This formula is handy for our exercise because we needed to find \( \sin(2\theta) \). By using \( \sin \theta = \frac{2\sqrt{6}}{5} \) and \( \cos \theta = \frac{1}{5} \), we plug in these values into the formula, resulting in \( \sin(2\theta) = \frac{4\sqrt{6}}{25} \).
Understanding double angle formulas facilitates working with trigonometric expressions, especially when transformations and simplifications involving angles are involved.