In Exercises 35–38, find the exact value of the following under the given conditions:βe. cos -------23 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Verified step by step guidance

1. Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \) and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).
2. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \alpha \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha \) is positive.
3. Calculate \( \cos \alpha \) using \( \cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2} \).
4. Use the Pythagorean identity to find \( \cos \beta \). Since \( \beta \) is in the second quadrant, \( \cos \beta \) is negative.
5. Calculate \( \cos \beta \) using \( \cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are the relationships between the angles and sides of a right triangle. The primary ratios include sine (sin), cosine (cos), and tangent (tan), which are defined as the ratios of the lengths of the sides opposite, adjacent, and hypotenuse, respectively. Understanding these ratios is essential for solving problems involving angles and lengths in trigonometry.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis. It is used to simplify the calculation of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. Knowing how to find and use reference angles is crucial for determining the exact values of trigonometric functions in various quadrants.

Quadrant Analysis

Quadrant analysis involves understanding the signs of trigonometric functions based on the quadrant in which the angle lies. The four quadrants are defined as follows: Quadrant I (all positive), Quadrant II (sin positive), Quadrant III (tan positive), and Quadrant IV (cos positive). This analysis is vital for determining the correct values of sine and cosine for angles in different ranges, especially when working with angles expressed in radians.