Solve each equation for exact solutions.
cos⁻¹ x = sin⁻¹ 3/5

Verified step by step guidance

1. Recognize that the equation is \(\cos^{-1} x = \sin^{-1} \frac{3}{5}\), where \(\cos^{-1} x\) and \(\sin^{-1} \frac{3}{5}\) represent inverse cosine and inverse sine functions respectively.
2. Recall the identity relating inverse sine and inverse cosine: for any angle \(\theta\), \(\cos^{-1} x = \sin^{-1} y\) implies \(x = \cos(\sin^{-1} y)\).
3. Use the Pythagorean identity to express \(\cos(\sin^{-1} y)\) in terms of \(y\): since \(\sin^2 \theta + \cos^2 \theta = 1\), then \(\cos(\sin^{-1} y) = \sqrt{1 - y^2}\) (considering the principal value range).
4. Substitute \(y = \frac{3}{5}\) into the expression to get \(x = \sqrt{1 - \left(\frac{3}{5}\right)^2}\).
5. Simplify the expression under the square root to find the exact value of \(x\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccos) and sin⁻¹ (arcsin), return the angle whose cosine or sine is a given value. They are used to find angles from known trigonometric ratios and have specific ranges to ensure unique outputs.

Relationship Between Sine and Cosine

Sine and cosine of complementary angles are related by the identity cos(θ) = sin(90° - θ) or cos(θ) = sin(π/2 - θ). This relationship helps in converting between inverse sine and inverse cosine expressions to find exact angle values.

Solving Trigonometric Equations for Exact Values

Solving trigonometric equations involves using known values of sine and cosine for special angles or applying identities to find exact solutions. Understanding the domain and range of inverse functions is essential to determine all valid solutions.