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. Recognize that the expression is \( \sin(\cos^{-1}(\frac{3}{5})) \). Here, \( \cos^{-1}(\frac{3}{5}) \) represents an angle \( \theta \) such that \( \cos(\theta) = \frac{3}{5} \).
2. Visualize or draw a right triangle where the adjacent side to angle \( \theta \) is 3 and the hypotenuse is 5, based on the cosine ratio \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
3. Use the Pythagorean theorem to find the length of the opposite side: \( \text{opposite} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} \).
4. Simplify the expression under the square root to find the opposite side length: \( \sqrt{16} \).
5. Calculate \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{16}}{5} \), which gives the exact value of \( \sin(\cos^{-1}(\frac{3}{5})) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like cos⁻¹(x), return the angle whose trigonometric ratio equals x. For example, cos⁻¹(3/5) gives the angle θ such that cos(θ) = 3/5. Understanding this allows us to interpret expressions involving inverse functions as angles.

Right Triangle Relationships

Using the value of cos(θ) = adjacent/hypotenuse, we can construct a right triangle with sides 3 (adjacent) and 5 (hypotenuse). The Pythagorean theorem helps find the opposite side, enabling us to find sin(θ) based on triangle side ratios.

Pythagorean Identity

The Pythagorean identity states sin²(θ) + cos²(θ) = 1. Given cos(θ), we can find sin(θ) by rearranging to sin(θ) = ±√(1 - cos²(θ)). This identity is essential for finding the sine of an angle when only the cosine is known.