Problem 27 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression. $$\sin \left(\cos ^{-1} \frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The exact value is \(\frac{4}{5}\).

Step 1: Understand the Problem

The expression involves finding the value of \( \sin(\cos^{-1} x) \) where \( x = \frac{3}{5} \). This requires understanding the inverse trigonometric function.

Step 2: Define Cosine Inverse

\( \cos^{-1} \frac{3}{5} \) is the angle whose cosine value is \( \frac{3}{5} \). Let's denote this angle by \( \theta \), so \( \cos \theta = \frac{3}{5} \).

Step 3: Use Pythagorean Identity

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). Substitute \( \cos \theta = \frac{3}{5} \) to get \( \sin^2 \theta = 1 - \left( \frac{3}{5} \right)^2 \).

Step 4: Calculate \(\sin^2 \theta\)

Calculate \( \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \).

Step 5: Determine \(\sin \theta\) Value

Since \( \sin^2 \theta = \frac{16}{25} \), then \( \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Step 6: Conclude the Exact Value

Hence, the exact value of \( \sin(\cos^{-1} \frac{3}{5}) \) is \( \frac{4}{5} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \(\theta\), the square of sine plus the square of cosine equals one. Mathematically, this identity is written as:

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

This identity is derived from the Pythagorean theorem and is crucial when working with trigonometric functions. It tells us how the sine and cosine of an angle relate to each other, forming a right triangle on the unit circle.
To solve problems like finding \(\sin(\cos^{-1} x)\), we use this identity to relate the known value of \(\cos \theta\) to \(\sin \theta\). For example, if \(\cos \theta = \frac{3}{5}\), the identity helps us find \(\sin \theta\) by rearranging it to \(\sin^2 \theta = 1 - \cos^2 \theta\). This manipulation allows us to solve for \(\sin \theta\) and find the exact values needed to solve trigonometric expressions.

Cosine Inverse

The Cosine Inverse function, denoted as \(\cos^{-1}(x)\), is one of the inverse trigonometric functions. It gives us the angle \(\theta\) whose cosine is the given value \(x\). In other words:

• If \(\cos \theta = x\), then \(\theta = \cos^{-1}(x)\)

This function is important because it reverses the action of the standard cosine function, allowing us to find angles from cosine values.
Understanding \(\cos^{-1}(x)\) is essential in solving trigonometric expressions, such as \(\sin(\cos^{-1} x)\). Here, \(\cos^{-1} \frac{3}{5}\) tells us the angle whose cosine is \(\frac{3}{5}\). This angle is critical because it allows us to move forward in calculations using trigonometric identities, like the Pythagorean Identity, to find related values like \(\sin \theta\). Recognizing the range and properties of inverse functions ensures accurate solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They simplify the computation of angles and lengths in trigonometry.
The most famous among these are the Pythagorean Identities:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

These identities help in swapping between trigonometric functions and finding unknown values. For instance, when given a cosine value in an inverse function context, you can use the Pythagorean Identity to find the equivalent sine value.
Knowing different identities enables us to simplify and solve complex trigonometric expressions. In our example, knowing that \(\cos \theta = \frac{3}{5}\) and using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we found \(\sin \theta\) as \(\frac{4}{5}\). Trigonometric identities are vital tools in mathematical problems involving angles and their trigonometric functions.