In Exercises 55–58, use the given information to find the exact value of each of the following:αb. cos ------2𝝅 sec α = ﹣3 , ------ < α < 𝝅2

Verified step by step guidance

1. Identify the given information: \( \sec \alpha = -3 \) and \( \frac{\pi}{2} < \alpha < \pi \). This indicates that \( \alpha \) is in the second quadrant.
2. Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \). Therefore, \( \cos \alpha = -\frac{1}{3} \).
3. Use the half-angle identity for cosine: \( \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}} \).
4. Substitute \( \cos \alpha = -\frac{1}{3} \) into the half-angle identity: \( \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \frac{1}{3}}{2}} \).
5. Simplify the expression under the square root and determine the correct sign for \( \cos \frac{\alpha}{2} \) based on the quadrant where \( \frac{\alpha}{2} \) lies.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as cosine (cos) and secant (sec), relate the angles of a triangle to the ratios of its sides. The secant function is defined as the reciprocal of the cosine function, meaning sec(α) = 1/cos(α). Understanding these functions is essential for solving problems involving angles and their corresponding values.

Quadrants and Angle Ranges

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle values. The given range for α, π/2 < α < π, indicates that α is in the second quadrant, where cosine values are negative. Recognizing the quadrant helps determine the sign of trigonometric functions based on the angle's location.

Exact Values of Trigonometric Functions

Finding exact values of trigonometric functions often involves using known values from special angles (like 30°, 45°, and 60°) or applying identities. In this case, since sec(α) = -3, we can find cos(α) by taking the reciprocal, leading to cos(α) = -1/3. Understanding how to manipulate these relationships is crucial for solving trigonometric equations.