Problem 109 - Precalculus Essentials

Question

In Exercises \(107-110,\) 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

\(\cos ^{2}\left(\frac{1}{2} \sin ^{-1} \frac{3}{5}\right) = \frac{16}{25}\)

Step 1: Find the angle

As \(\sin ^{-1} \frac{3}{5}\) represents an angle whose sine value is 3/5, this angle can be referred to as \(x\). Therefore, we have \(\sin(x) = \frac{3}{5}\)

Step 2: Use Pythagorean identity

Now, we can use the Pythagorean identity which states \(\sin^2(x) + \cos^2(x) = 1\). Substituting the value of \(\sin(x)\) into this equation, we get \( (\frac{3}{5})^2 + \cos^2(x) = 1\). Solving this equation, we obtain \(\cos^2(x) = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}\)

Step 3: Substitute the angle

Again, replace \(x\) with \(\frac{1}{2} \sin ^{-1} \frac{3}{5}\) to get the answer

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial in mathematics as they allow us to determine the angle when given the value of the trigonometric function. In this context, \(\sin^{-1} \frac{3}{5}\) is an inverse function that tells us at what angle, denoted by \(x\), the sine value equals \(3/5\). In simple terms, it is the 'undoing' of the sine function to find the angle from a specific ratio. When working with these functions, remember that they have restricted ranges. For instance:

• \(\sin^{-1}(x)\) returns angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• Each inverse trigonometric function gives a particular set of angle values depending on its definition.

This restricted range assures that for any input value, there is a single and unique angle output, which is very important for defining functions in mathematics. In many problems, like the one at hand, inverse trigonometric functions help convert a ratio back into an angle, which can be used with other identities to simplify or solve expressions.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, expressing the relation between sine and cosine for any angle. It is given by the equation:\[\sin^2(x) + \cos^2(x) = 1\]This identity is vital because it helps find one trigonometric function value if the other is known. In the exercise provided, knowing \(\sin(x) = \frac{3}{5}\) enables us to compute \(\cos(x)\) using:

• Substitute \(\sin(x) = \frac{3}{5}\).
• Square the sine value to get \(\left(\frac{3}{5}\right)^2 = \frac{9}{25}\).
• Plug this result into the Pythagorean identity: \(\sin^2(x) + \cos^2(x) = 1\).

After substituting, the equation becomes \(\frac{9}{25} + \cos^2(x) = 1\), and solving it gives \(\cos^2(x) = \frac{16}{25}\). This identity hence provides an invaluable tool when handling expressions involving squares of sine and cosine functions.

Sine and Cosine Relations

In trigonometry, sine and cosine are closely related as they are both fundamental trigonometric functions depicting ratios in right-angled triangles:

• \(\sin(x)\) represents the ratio of the opposite side to the hypotenuse.
• \(\cos(x)\) represents the ratio of the adjacent side to the hypotenuse.

These two functions are related through the complementary angles. Specifically, \(\cos(90^\circ - x) = \sin(x)\) and \(\sin(90^\circ - x) = \cos(x)\). However, beyond these complements, the relationship:\[\sin^2(x) + \cos^2(x) = 1\]forms the backbone of connecting sine and cosine together.In the exercise, the half-angle theorem or substitution \(x = \frac{1}{2} \sin^{-1} \rac{3}{5}\) might be implied, where angles are halved and subsequently worked into trigonometric identities. This strategy simplifies complex expressions into something more manageable through these core trigonometric relationships. Hence, understanding how sine and cosine integrate through various identities is essential for mastering trigonometry.