In Exercises 55–58, use the given information to find the exact value of each of the following:αb. cos ------24tan α = ------ , 180° < α < 270°3

Verified step by step guidance

1. insert step 1: Recognize that the given angle \( \alpha \) is in the third quadrant, where both sine and cosine are negative.
2. insert step 2: Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to express \( \sin \alpha \) and \( \cos \alpha \) in terms of each other. Given \( \tan \alpha = \frac{4}{3} \), assume \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).
3. insert step 3: Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \). Substitute \( (-4k)^2 + (-3k)^2 = 1 \) and solve for \( k \).
4. insert step 4: Once \( k \) is found, determine \( \sin \alpha \) and \( \cos \alpha \) using \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).
5. insert step 5: Use the half-angle identity for cosine: \( \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}} \). Since \( \alpha \) is in the third quadrant, \( \frac{\alpha}{2} \) is in the second quadrant, where cosine is negative. Calculate \( \cos \frac{\alpha}{2} \) using the values found for \( \cos \alpha \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this context, cosine and tangent are particularly important for finding the values of angles in different quadrants. Understanding how these functions behave in various quadrants is essential for solving the problem.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific angle ranges. For angles between 180° and 270°, the angle is located in the third quadrant, where both sine and cosine values are negative, while tangent values are positive. Recognizing the quadrant helps determine the signs of the trigonometric functions involved.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions can be derived from special angles (like 30°, 45°, and 60°) or using the Pythagorean identity. In this problem, knowing how to manipulate the given tangent value to find cosine involves using the identity tan(α) = sin(α)/cos(α) and the relationship between sine and cosine in the context of the unit circle.