9.3 Double-angle, half-angle, and reduction formulas  (Page 4/8)

Explain how to determine the reduction identities from the double-angle identity $\cos \left(2x\right)={\cos }^{2}x-{\sin }^{2}x.$

Explain how to determine the double-angle formula for $\tan (2x)$ using the double-angle formulas for $\cos (2x)$ and $\sin (2x).$

We can determine the half-angle formula for $\tan \left(\frac{x}{2}\right)=\frac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}$ by dividing the formula for $\sin \left(\frac{x}{2}\right)$ by $\cos \left(\frac{x}{2}\right).$ Explain how to determine two formulas for $\tan \left(\frac{x}{2}\right)$ that do not involve any square roots.

For the half-angle formula given in the previous exercise for $\tan \left(\frac{x}{2}\right),$ explain why dividing by 0 is not a concern. (Hint: examine the values of $\cos x$ necessary for the denominator to be 0.)

If $\sin x=\frac{1}{8},$ and $x$ is in quadrant I.

If $\cos x=\frac{2}{3},$ and $x$ is in quadrant I.

If $\cos x=-\frac{1}{2},$ and $x$ is in quadrant III.