Powers of Functions — Hard Practice Quiz

A Trigonometry cheat sheet for Powers of Functions — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. Express $\sin^4 \alpha$ in terms of $\cos(2\alpha)$ and $\cos(4\alpha)$ using the provided formulas.

   • $\frac{3 - 4\cos(2\alpha) + \cos(4\alpha)}{8}$
   • $\frac{3 + 4\cos(2\alpha) + \cos(4\alpha)}{8}$
   • $\frac{1 - 2\cos(2\alpha) + \cos(4\alpha)}{4}$
   • $\frac{1 - 2\cos(2\alpha) + \cos^2(2\alpha)}{4}$

   Answer: $\frac{3 - 4\cos(2\alpha) + \cos(4\alpha)}{8}$

2. Simplify the expression $\sin^2 \alpha \cos^2 \alpha$ in terms of $\cos(4\alpha)$ using the provided formulas.

   • $\frac{1 - \cos(4\alpha)}{8}$
   • $\frac{1 + \cos(4\alpha)}{8}$
   • $\frac{1 - \cos^2(2\alpha)}{4}$
   • $\frac{\sin^2(2\alpha)}{4}$

   Answer: $\frac{1 - \cos(4\alpha)}{8}$

3. Algebraically manipulate the given formula for $\sin^3 \alpha$ to express $\sin(3\alpha)$ in terms of $\sin \alpha$ and its powers.

   • $3\sin \alpha - 4\sin^3 \alpha$
   • $4\sin^3 \alpha - 3\sin \alpha$
   • $\frac{3\sin \alpha - \sin(3\alpha)}{4}$
   • $3\sin \alpha + 4\sin^3 \alpha$

   Answer: $3\sin \alpha - 4\sin^3 \alpha$

4. Algebraically manipulate the given formula for $\cos^3 \alpha$ to express $\cos(3\alpha)$ in terms of $\cos \alpha$ and its powers.

   • $4\cos^3 \alpha - 3\cos \alpha$
   • $3\cos \alpha - 4\cos^3 \alpha$
   • $\frac{3\cos \alpha + \cos(3\alpha)}{4}$
   • $4\cos^3 \alpha + 3\cos \alpha$

   Answer: $4\cos^3 \alpha - 3\cos \alpha$

5. Simplify the sum $\sin^3 \alpha + \cos^3 \alpha$ using the provided formulas, expressing the result in terms of $\sin \alpha$, $\cos \alpha$, $\sin(3\alpha)$, and $\cos(3\alpha)$.

   • $\frac{3(\sin \alpha + \cos \alpha) - (\sin(3\alpha) - \cos(3\alpha))}{4}$
   • $\frac{3(\sin \alpha + \cos \alpha) + (\sin(3\alpha) + \cos(3\alpha))}{4}$
   • $\frac{3\sin \alpha - \sin(3\alpha) + 3\cos \alpha + \cos(3\alpha)}{4}$
   • $\frac{3(\sin \alpha - \cos \alpha) - (\sin(3\alpha) + \cos(3\alpha))}{4}$

   Answer: $\frac{3(\sin \alpha + \cos \alpha) - (\sin(3\alpha) - \cos(3\alpha))}{4}$

6. If $\cos(2\alpha) = x$, express $\sin^2 \alpha \cos^2 \alpha$ in terms of $x$ using the provided formulas and other relevant trigonometric identities.

   • $\frac{1 - x^2}{4}$
   • $\frac{1 + x^2}{4}$
   • $\frac{1 - 2x^2}{8}$
   • $\frac{1 - x^2}{8}$

   Answer: $\frac{1 - x^2}{4}$

7. Using the provided formulas, evaluate the expression $\sin^2(\frac{\pi}{8}) + \cos^2(\frac{\pi}{8})$.

   • $1$
   • $\frac{1 + \cos(\frac{\pi}{4})}{2}$
   • $\frac{1 - \cos(\frac{\pi}{4})}{2}$
   • $\cos(\frac{\pi}{4})$

   Answer: $1$

8. Express the ratio $\frac{\sin^2 \alpha}{\cos^2 \alpha}$ in terms of $\cos(2\alpha)$ using the provided formulas.

   • $\frac{1 - \cos(2\alpha)}{1 + \cos(2\alpha)}$
   • $\frac{1 + \cos(2\alpha)}{1 - \cos(2\alpha)}$
   • $\frac{\sin(2\alpha)}{\cos(2\alpha)}$
   • $\frac{1 - \cos^2(2\alpha)}{1 + \cos^2(2\alpha)}$

   Answer: $\frac{1 - \cos(2\alpha)}{1 + \cos(2\alpha)}$

9. Given $\sin(3\alpha) = A$ and $\cos(3\alpha) = B$, express $\sin^3 \alpha - \cos^3 \alpha$ in terms of $A$, $B$, $\sin \alpha$, and $\cos \alpha$ using the provided formulas.

   • $\frac{3(\sin \alpha - \cos \alpha) - (A + B)}{4}$
   • $\frac{3(\sin \alpha + \cos \alpha) - (A - B)}{4}$
   • $\frac{3\sin \alpha - A - 3\cos \alpha + B}{4}$
   • $\frac{3\sin \alpha - A + 3\cos \alpha + B}{4}$

   Answer: $\frac{3(\sin \alpha - \cos \alpha) - (A + B)}{4}$

10. Given $\sin^2 \alpha = \frac{1}{3}$, find the exact value of $\cos(4\alpha)$ using the provided formulas.

    • $-\frac{7}{9}$
    • $\frac{7}{9}$
    • $\frac{1}{3}$
    • $-\frac{1}{3}$

    Answer: $-\frac{7}{9}$