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

Verified step by step guidance

1. insert step 1: Recognize that the 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 a common variable.
3. insert step 3: Given \( \tan \alpha = \frac{4}{3} \), let \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \) for some positive \( k \).
4. insert step 4: Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \).
5. insert step 5: Use the double angle identity for sine, \( \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}} \), to find \( \sin \frac{\alpha}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and tangent, relate the angles of a triangle to the ratios of its sides. For example, the sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse, while the tangent is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for solving problems involving angles and their corresponding side lengths.

Quadrants of the Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is divided into four quadrants, each corresponding to different signs of the sine and cosine values. In the third quadrant (180° < α < 270°), both sine and cosine are negative, which affects the values of trigonometric functions and is crucial for determining the correct signs when calculating values.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to find the values of trigonometric functions for angles greater than 90° or less than 0°. For angles in the third quadrant, the reference angle can help determine the sine and tangent values by relating them back to their corresponding acute angles, allowing for accurate calculations of trigonometric functions.