2nd Hour Pre-Calculus B (Spring 2012): Power-Reducing Formulas, Half-Angle Formulas, Sum-to-Product Formulas, Product-to-Sum Formulas

Power-Reducing Formulas

How to derive the $\sin ^2\Theta$ power-reducing formula:

Start out with an formula that includes both cosine and sine

$\cos 2\Theta = 1-2sin^2\Theta$
Solve for $\sin ^2\Theta$

First add $2\sin ^2\Theta$ to each side. This will cancel it on the right side of the formula. Then you must subtract $\cos2\Theta$ from each side. Your formula should now look like this:

$2\sin ^2\Theta = 1-cos2\Theta$

Now divide each side by 2 and you will get the sine power-reducing formula.

Half-Angle Formulas

How to derive the sine half-angle formula:

$\sin ^2\Theta =\frac{1-\cos 2\Theta }{2}$
$u=2\Theta$
This allows you to substitute u into the equation for theta.

Now your formula should look like:

$\sin ^2\frac{u}{2}=\frac{1-\cos u}{2}$
Take the square root of each side. Remember that the square root of the right side will be $\pm$
The formula should now look like this:

$sin\frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}$
Your final answer can either be + or - but it can not be both! The sign of the final answer depends where it is located on the unit circle.

If you wanted to find $\sin \frac{\pi }{8}$ you would have to use $\frac{\pi }{4}$ in the half-angle formula.

Sum-to-Product Formulas

Product-to-Sum Formulas

2nd Hour Pre-Calculus B (Spring 2012)

Thursday, April 5, 2012

Power-Reducing Formulas, Half-Angle Formulas, Sum-to-Product Formulas, Product-to-Sum Formulas

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