9.3 Double-Angle, Half-Angle, and Reduction Formulas

Given $\sin \alpha = \frac{5}{8} ,$ with $\alpha$ in quadrant I, find $\cos \left(2 \alpha\right) .$

Use the double-angle formula for cosine to write $\cos \left(6 x\right)$ in terms of $\cos \left(3 x\right) .$

Verify the following identity using double-angle formulas:

Verify the identity: $\cos^{4} \theta - \sin^{4} \theta = \cos \left(2 \theta\right) .$

Verify the identity:

Verify the identity: $\cos \left(2 \theta\right) \cos \theta = \cos^{3} \theta - \cos \theta \sin^{2} \theta .$

Write an equivalent expression for $\cos^{4} x$ that does not involve any powers of sine or cosine greater than 1.

Use the power-reducing formulas to prove

Use the power-reducing formulas to prove that $10 \cos^{4} x = \frac{15}{4} + 5 \cos \left(2 x\right) + \frac{5}{4} \cos \left(4 x\right) .$

Find $\sin \left(15°\right)$ using a half-angle formula.

Given that $\sin \alpha = - \frac{4}{5}$ and $\alpha$ lies in quadrant IV, find the exact value of $\cos \left(\frac{\alpha}{2}\right) .$

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

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.

Find the exact values of a) $\sin \left(2 x\right) ,$ b) $\cos \left(2 x\right) ,$ and c) $\tan \left(2 x\right)$ without solving for $x .$

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

Find the exact values of a) $\sin \left(2 x\right) ,$ b) $\cos \left(2 x\right) ,$ and c) $\tan \left(2 x\right)$ without solving for $x .$

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

Find the values of the six trigonometric functions if the conditions provided hold.

$\cos \left(2 \theta\right) = \frac{3}{5}$ and $90° \leq \theta \leq 180°$

Simplify to one trigonometric expression.

$2 \sin \left(\frac{\pi}{4}\right) \cos \left(\frac{\pi}{4}\right)$

Find the exact value using half-angle formulas.

$\sin \left(\frac{\pi}{8}\right)$

Find the exact value using half-angle formulas.

$\sin \left(\frac{11 \pi}{12}\right)$

Find the exact value using half-angle formulas.

$\tan \left(\frac{5 \pi}{12}\right)$

Find the exact value using half-angle formulas.

$\tan \left(- \frac{3 \pi}{8}\right)$

Find the exact values of a) $\sin \left(\frac{x}{2}\right) ,$ b) $\cos \left(\frac{x}{2}\right) ,$ and c) $\tan \left(\frac{x}{2}\right)$ without solving for $x ,$ when $0 \leq x < 2 \pi$

If $\sin x = - \frac{12}{13} ,$ and $x$ is in quadrant III.

Find the exact values of a) $\sin \left(\frac{x}{2}\right) ,$ b) $\cos \left(\frac{x}{2}\right) ,$ and c) $\tan \left(\frac{x}{2}\right)$ without solving for $x ,$ when $0 \leq x < 2 \pi$

If $\sec x = - 4 ,$ and $x$ is in quadrant II.

Find $\sin \left(2 \alpha\right) , \cos \left(2 \alpha\right) ,$ and $\tan \left(\right. 2 \alpha ).$

Find $\sin \left(\frac{\alpha}{2}\right) , \cos \left(\frac{\alpha}{2}\right) ,$ and $\tan \left(\frac{\alpha}{2}\right) .$

Simplify each expression. Do not evaluate.

$2 \cos^{2} \left(37°\right) - 1$

Simplify each expression. Do not evaluate.

$\cos^{2} \left(9 x\right) - \sin^{2} \left(9 x\right)$

Simplify each expression. Do not evaluate.

$6 \sin \left(5 x\right) \cos \left(5 x\right)$

Prove the given identity.

$\sin \left(2 x\right) = - 2 \sin \left(- x\right) \cos \left(- x\right)$

Prove the given identity.

$\frac{\sin \left(2 \theta\right)}{1 + \cos \left(2 \theta\right)} \tan^{2} \theta = \tan^{3} \theta$

Rewrite the expression with an exponent no higher than 1.

$\cos^{2} \left(6 x\right)$

Rewrite the expression with an exponent no higher than 1.

$\sin^{4} \left(3 x\right)$

Rewrite the expression with an exponent no higher than 1.

$\cos^{4} x \left( \sin\right)^{2} x$

Reduce the equations to powers of one, and then check the answer graphically.

$\tan^{4} x$

Reduce the equations to powers of one, and then check the answer graphically.

$\sin^{2} x \left( \cos\right)^{2} x$

Reduce the equations to powers of one, and then check the answer graphically.

$\tan^{4} x \left( \cos\right)^{2} x$

Reduce the equations to powers of one, and then check the answer graphically.

$\cos^{2} \left(2 x\right) \sin x$

Algebraically find an equivalent function, only in terms of $\sin x$ and/or $\cos x ,$ and then check the answer by graphing both functions.

$\sin \left(4 x\right)$

Prove the identities.

$\sin \left(2 x\right) = \frac{2 \tan x}{1 + \tan^{2} x}$

Prove the identities.

$\tan \left(2 x\right) = \frac{2 \sin x \cos x}{2 \cos^{2} x - 1}$

Prove the identities.

$\sin \left(3 x\right) = 3 \sin x \cos^{2} x - \sin^{3} x$

Prove the identities.

$\frac{1 + \cos \left(2 t\right)}{\sin \left(2 t\right) - \cos t} = \frac{2 \cos t}{2 \sin t - 1}$

Prove the identities.

$\cos \left(16 x\right) = \left(\cos^{2} \left(4 x\right) - \sin^{2} \left(4 x\right) - \sin \left(8 x\right)\right) \left(\cos^{2} \left(4 x\right) - \sin^{2} \left(4 x\right) + \sin \left(8 x\right)\right)$