In Exercises 29–51, find the exact value of each expression. Do not use a calculator.sin(cos⁻¹ 3/5)

Verified step by step guidance

1. insert step 1: Understand that you need to find \( \sin(\theta) \) where \( \theta = \cos^{-1}(3/5) \). This means \( \cos(\theta) = 3/5 \).
2. insert step 2: Use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find \( \sin(\theta) \).
3. insert step 3: Substitute \( \cos(\theta) = 3/5 \) into the identity: \( \sin^2(\theta) + (3/5)^2 = 1 \).
4. insert step 4: Simplify the equation: \( \sin^2(\theta) + 9/25 = 1 \).
5. insert step 5: Solve for \( \sin^2(\theta) \) and then take the square root to find \( \sin(\theta) \). Remember to consider the range of \( \theta \) to determine the sign of \( \sin(\theta) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find the angle whose cosine is a given value. In this case, cos⁻¹(3/5) gives an angle θ such that cos(θ) = 3/5. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity allows us to find the sine of an angle if we know its cosine. In this problem, knowing cos(θ) = 3/5 enables us to calculate sin(θ) using the identity, which is essential for finding sin(cos⁻¹(3/5)).

Trigonometric Ratios in Right Triangles

Trigonometric ratios relate the angles and sides of right triangles. For an angle θ, sin(θ) is defined as the ratio of the length of the opposite side to the hypotenuse. By visualizing the triangle formed by the angle θ where cos(θ) = 3/5, we can determine the lengths of the sides and subsequently find sin(θ), which is necessary for solving the given expression.