Problem 109 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ \cos ^{2}\left(\frac{1}{2} \sin ^{-1} \frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The exact value of the given expression is \( \frac{9}{10} \)

Step 1: Find the angle whose sine value is known

Here we have \( \frac{1}{2} sin^{-1} \frac{3}{5} \). This represents an angle whose sine value is \(\frac{3}{5}\). First, consider an angle \( \alpha \) such that \( \sin(\alpha) = \frac{3}{5} \)

Step 2: Use the Pythagorean identity to find cosine of the angle

The cosine of an angle can be found using the Pythagorean identity which states \( \cos^2(\alpha) = 1 - \sin^2(\alpha) \). In this case, our \(\alpha\) has a sine of \(\frac{3}{5}\). Plugging in we have \( \cos^2(\alpha) = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \). Therefore, \( \cos(\alpha) = \frac{4}{5} \) or \( -\frac{4}{5} \). We choose the positive root because \( \alpha \) is in the first quadrant, where cosine is positive.

Step 3: Find \(\cos^2\left(\frac{1}{2}\alpha\right)\)

The objective is now to find the square of the cosine of half the angle. From half-angle formula, we know \( \cos^2\left(\frac{1}{2}\alpha\right) = \frac{1+\cos(\alpha)}{2} \). Substituting the value of \( \cos(\alpha) \) from Step 2, we get \( \cos^2\left(\frac{1}{2}\alpha\right) = \frac{1 + \frac{4}{5}}{2} = \frac{9}{10} \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is an essential part of trigonometry, expressing a fundamental relationship between the sine and cosine of an angle. It states that for any angle \( \theta \), the square of the sine plus the square of the cosine equals one: \[ \sin^{2}(\theta) + \cos^{2}(\theta) = 1 \. \] This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. For our exercise, the knowledge of this identity allows us to find the cosine of an angle if we already know the sine.

After we established that \( \sin(\alpha) = \frac{3}{5} \) for a certain angle \( \alpha \), we use the Pythagorean identity to find out that \( \cos^{2}(\alpha) = 1 - \sin^{2}(\alpha) = \frac{16}{25} \) and subsequently determine that \( \cos(\alpha) = \frac{4}{5} \) because it is in the first quadrant where cosine values are positive.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backwards in trigonometry. When we know the ratio of sides in a right triangle (such as the opposite side to the hypotenuse for sine), we can find the angle that produces this ratio. These functions are denoted with the 'inverse' notation, like \( \sin^{-1} \) for arcsine.

These functions have ranges that are principal values for each ratio to ensure each function is one-to-one. In the exercise, we begin by finding the angle whose sine is \( \frac{3}{5} \) using \( \sin^{-1} \) or arcsine. The value of this angle is then used to determine the cosine of half the angle, which requires additional trigonometric identities.

Half-Angle Formulas

Half-angle formulas are trigonometric identities that relate the trigonometric functions of half-angles to the functions of the full angle. These are particularly useful when dealing with problems where we know the trigonometric function of the full angle and we need to find the function of the half angle.

The formula for cosine is given by \[ \cos^{2}\left(\frac{1}{2}\theta\right) = \frac{1+\cos(\theta)}{2} \.\] In our case, after we determine \( \cos(\alpha) = \frac{4}{5} \) using the Pythagorean identity, we apply the half-angle formula to find \( \cos^{2}\left(\frac{1}{2}\alpha\right) = \frac{1 + \frac{4}{5}}{2} = \frac{9}{10} \.\)

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. They enable us to simplify and manipulate expressions and solve trigonometric equations. Common identities include the reciprocal identities, quotient identities, Pythagorean identities, co-function identities, and of course, the half-angle formulas mentioned previously.

These identities are tools in our trigonometric toolkit that allow us to find unknown angles, simplify expressions, and solve for values without having to measure them each time. In the step by step solution of the exercise, we made use of these identities to progress from knowing the sine of an angle to finding the square of the cosine of half that angle.