Question

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(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 $$ 	an \left( \frac{	heta}{2} ight) $$, the tangent of half the angle $$ 	heta . $$Recall the half-angle identity for tangent:

$$ 	an \left( \frac{	heta}{2} ight) = \pm \sqrt{\frac{1 - \cos 	heta}{1 + \cos 	heta}}

or $$alternatively,

$$ 	an \left( \frac{	heta}{2} ight) = \frac{\sin 	heta}{1 + \cos 	heta} \quad 	ext{or} \quad 	an \left( \frac{	heta}{2} ight) = \frac{1 - \cos 	heta}{\sin 	heta} $$

Choose the form that best fits the information given in the figure. Determine the values of $$ \sin 	heta $$ and $$ \cos 	heta $$ from the figure or from the problem data. This might involve using the coordinates of a point on the unit circle, or lengths of sides in a right triangle related to $$ 	heta . $$Substitute the values of $$ \sin 	heta $$ and $$ \cos 	heta $$ into the chosen half-angle formula for $$ 	an \left( \frac{	heta}{2} ight) . $$Simplify the expression carefully, paying attention to the sign of the tangent function in the quadrant where $$ \frac{	heta}{2} $$ lies, to find the exact value of $$ 	an \left( \frac{	heta}{2} ight) .$$

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

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Key Concepts

Definition of the Tangent Function

Using Figures to Determine Side Lengths

Exact Values of Trigonometric Functions