In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°

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Identify the quadrant where the angle \( \alpha \) lies. Since \( 180^\circ < \alpha < 270^\circ \), \( \alpha \) is in the third quadrant, where both sine and cosine values are negative, but tangent is positive.

Recall the given value: \( \tan \alpha = \frac{4}{3} \). Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to relate sine and cosine.

Set \( \sin \alpha = 4k \) and \( \cos \alpha = 3k \) for some constant \( k \), because the ratio of sine to cosine must be \( \frac{4}{3} \). Since \( \alpha \) is in the third quadrant, both sine and cosine are negative, so \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).

Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \): \[ (-4k)^2 + (-3k)^2 = 1 \\ 16k^2 + 9k^2 = 1 \\ 25k^2 = 1 \\ k^2 = \frac{1}{25} \\ k = \frac{1}{5} \]

Substitute \( k = \frac{1}{5} \) back into \( \sin \alpha = -4k \) to find \( \sin \alpha \): \[ \sin \alpha = -4 \times \frac{1}{5} = -\frac{4}{5} \]

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Trigonometric Ratios and Their Signs in Quadrants

Trigonometric functions like sine and tangent have specific signs depending on the quadrant of the angle. Since 180° < α < 270°, α lies in the third quadrant where sine is negative and tangent is positive. Understanding these sign rules is essential to determine the correct value of sin(α/2).

Half-Angle Formulas

Half-angle formulas allow calculation of trigonometric values for half of a given angle. For sine, the formula is sin(α/2) = ±√((1 - cos α)/2). Choosing the correct sign depends on the quadrant of α/2, which must be determined from the original angle α.

Relationship Between Tangent and Cosine

Given tan α, you can find cos α using the identity tan α = sin α / cos α and the Pythagorean identity sin² α + cos² α = 1. This step is crucial because the half-angle formula for sine requires the value of cos α.

In Exercises 55–58, use the given information to find the exact value of each of the following: a. sin(α/2)tan α = 4/3, 180° < α < 270°