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

Verified step by step guidance

1. Identify the quadrant in which angle \( \alpha \) lies. Since \( \frac{\pi}{2} < \alpha < \pi \), \alpha \) is in the second quadrant.
2. Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \). Given \( \sec \alpha = -3 \), we have \( \cos \alpha = -\frac{1}{3} \).
3. Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to find \( \sin \alpha \). Substitute \( \cos \alpha = -\frac{1}{3} \) into the identity.
4. Solve for \( \sin \alpha \) and remember that in the second quadrant, \( \sin \alpha \) is positive.
5. Use the identity for tangent of half-angle: \( \tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} \). Substitute \( \cos \alpha = -\frac{1}{3} \) into this formula to find \( \tan \frac{\alpha}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, secant, and others, relate the angles of a triangle to the lengths of its sides. Understanding these functions is crucial for solving problems involving angles and their relationships. For example, the tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle.

Secant Function

The secant function is the reciprocal of the cosine function, defined as sec(α) = 1/cos(α). It is important to know how to manipulate and interpret secant values, especially when given in a problem. In this case, sec(α) = -3 indicates that cos(α) = -1/3, which helps in determining the angle α within the specified range.

Angle Ranges and Quadrants

Understanding the range of angles and their corresponding quadrants is essential in trigonometry. The given range for α, π/2 < α < π, indicates that α is in the second quadrant, where sine is positive and cosine is negative. This knowledge helps in determining the signs of the trigonometric functions and finding the exact values of tan(α/2) using the half-angle identities.