Question

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

Accepted Answer

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

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

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Key Concepts

Trigonometric Functions

Unit Circle and Angle Measures

Exact Values of Common Angles