In Exercises 35–38, find the exact value of the following under the given conditions:b. cos(α﹣β)3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Verified step by step guidance

1. Step 1: Use the identity for cos(\(\alpha - \beta\)): \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).
2. Step 2: Find \(\cos \alpha\) using \(\sin \alpha = \frac{3}{5}\). Since \(0 < \alpha < \frac{\pi}{2}\), \(\cos \alpha = \sqrt{1 - \sin^2 \alpha}\).
3. Step 3: Find \(\cos \beta\) using \(\sin \beta = \frac{12}{13}\). Since \(\frac{\pi}{2} < \beta < \pi\), \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\).
4. Step 4: Substitute \(\cos \alpha\), \(\cos \beta\), \(\sin \alpha\), and \(\sin \beta\) into the identity from Step 1.
5. Step 5: Simplify the expression to find the exact value of \(\cos(\alpha - \beta)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The cosine of the difference of two angles, cos(α - β), is expressed as cos(α)cos(β) + sin(α)sin(β). Understanding these identities is crucial for simplifying expressions and solving problems in trigonometry.

Sine and Cosine Values

To find cos(α - β), we need the sine and cosine values of angles α and β. Given sin(α) and sin(β), we can derive cos(α) and cos(β) using the Pythagorean identity: cos²(θ) + sin²(θ) = 1. This relationship allows us to calculate the cosine values necessary for applying the cosine difference identity.

Angle Restrictions

The problem specifies ranges for angles α and β, which are important for determining the signs of the sine and cosine values. For 0 < α < 3π/2, α is in the first or second quadrant, where sine is positive and cosine can be negative. For 2π/13 < β < π, β is in the second quadrant, where sine is positive and cosine is negative. These restrictions affect the final value of cos(α - β).