In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(θ/2)

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Identify the given angle \( \theta \) and the trigonometric function you need to find. In this case, it appears you need to find \( \sin \left( \frac{\theta}{2} \right) \), which is the sine of half the angle \( \theta \).

Recall the half-angle identity for sine: \[ \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{ \frac{1 - \cos \theta}{2} } \] This formula allows you to find the sine of half an angle if you know \( \cos \theta \).

Determine the sign (positive or negative) of \( \sin \left( \frac{\theta}{2} \right) \) based on the quadrant where \( \frac{\theta}{2} \) lies. Remember that sine is positive in the first and second quadrants and negative in the third and fourth quadrants.

Find or calculate \( \cos \theta \) from the given figure or information. This might involve using the Pythagorean theorem if the figure provides side lengths, or using other trigonometric relationships.

Substitute the value of \( \cos \theta \) into the half-angle formula and simplify the expression under the square root. This will give you the exact value of \( \sin \left( \frac{\theta}{2} \right) \) up to the sign determined earlier.

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