Half Angle Formulas

Definition: Half angle formulas are identities which can be used to find exact values in trigonometric expressions:

Example: Given cos x = -15/17 and 180 <x < 270, find tan (x/2).

Solution:

Step 1: Looking at the formula for tan (x/2), we can see cos x will be needed to solve. Plug cos x = -15/17 into the formula. Because x is in Quadrant III where tan is positive, the plug or minus sign becomes positive.$\dpi{150} \bg_white \\tan\frac{x}{2} = \pm \sqrt{\frac{1-cosx}{1+cosx}}\\ \\tan\frac{x}{2} = \sqrt{\frac{1+(15/17)}{1-(15/17)}}\\$Step 2: Simplify.

$\dpi{150} \bg_white \\tan\frac{x}{2} = \sqrt{\frac{1+(15/17)}{1-(15/17)}}\\ \\tan\frac{x}{2} = \sqrt{\frac{32/17}{2/17}}\\ \\tan\frac{x}{2} = \sqrt{16} = 4$

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