In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

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Identify the given information: \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = -\frac{1}{3} \) with \( \pi < \beta < \frac{3\pi}{2} \). Both angles are in the third quadrant.

Recall that in the third quadrant, both sine and cosine values are negative, which is consistent with the given values and intervals.

Use the Pythagorean identity to find \( \cos \alpha \): \( \cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \). The negative sign is chosen because cosine is negative in the third quadrant.

Similarly, find \( \sin \beta \) using the Pythagorean identity: \( \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \), since sine is also negative in the third quadrant.

To find \( \cos \frac{\beta}{2} \), use the half-angle formula: \[ \cos \frac{\beta}{2} = \pm \sqrt{\frac{1 + \cos \beta}{2}}. \] Determine the correct sign based on the quadrant where \( \frac{\beta}{2} \) lies, considering \( \pi < \beta < \frac{3\pi}{2} \).

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Trigonometric Ratios and Their Signs in Different Quadrants

Trigonometric functions like sine and cosine have specific signs depending on the quadrant of the angle. For example, sine is positive in the first and second quadrants, while cosine is positive in the first and fourth. Understanding the given interval for angles α and β helps determine the correct sign of the trigonometric values.

Exact Values of Trigonometric Functions

Exact values refer to precise trigonometric ratios often derived from special angles such as π/3, π/2, etc. These values are expressed in simplified radical form or fractions, not decimals. Recognizing these angles and their sine or cosine values is essential for solving problems without approximation.

Using Trigonometric Identities to Find Unknown Values

Identities like sin²θ + cos²θ = 1 allow calculation of unknown trigonometric values when one is given. This is crucial when the problem provides sine or cosine and requires finding the other function or related expressions, ensuring the solution respects the angle's quadrant.

In Exercises 35–38, find the exact value of the following under the given conditions:e. cos( β/2)sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.