Multiple Choice

1. Solve for all real $x$: $\sin x=\tfrac{1}{2}$.

   $x=\pm\tfrac{\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{2\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+\pi n$ or $x=\tfrac{5\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

2. Solve for all real $x$: $\cos x=\tfrac{\sqrt3}{2}$.

   $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

3. Solve for all real $x$: $\sin x=-\tfrac{\sqrt2}{2}$.

   $x=\pm\tfrac{\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{5\pi}{4}+\pi n$ or $x=\tfrac{7\pi}{4}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{3\pi}{4}+2\pi n$ or $x=\tfrac{5\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{5\pi}{4}+2\pi n$ or $x=\tfrac{7\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

4. Solve for all real $x$: $\cos x=-\tfrac{1}{2}$.

   $x=\tfrac{2\pi}{3}+\pi n$ or $x=\tfrac{4\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{5\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{2\pi}{3}+2\pi n$ or $x=\tfrac{4\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

5. Solve for all real $x$: $2\sin x=1$.

   $x=\pm\tfrac{\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{2\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+\pi n$ or $x=\tfrac{5\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

6. Solve for all real $x$: $2\cos x=\sqrt3$.

   $x=\pm\tfrac{\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

7. Solve for all real $x$: $\sin(2x)=\tfrac{\sqrt3}{2}$.

   $x=\tfrac{\pi}{6}+\pi n$ or $x=\tfrac{\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{12}+\pi n$ or $x=\tfrac{5\pi}{12}+\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

8. Solve for all real $x$: $\cos(2x)=-\tfrac{1}{2}$.

   $x=\tfrac{2\pi}{3}+2\pi n$ or $x=\tfrac{4\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{3}+\pi n$ or $x=\tfrac{2\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

   $x=\pm\tfrac{\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{2\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

9. Solve for all real $x$: $\sin(3x)=0$.

   $x=\tfrac{\pi}{3}n,\ n\in\mathbb{Z}$.

   $x=\tfrac{\pi}{2}n,\ n\in\mathbb{Z}$.

   $x=\tfrac{2\pi}{3}n,\ n\in\mathbb{Z}$.

   $x=\pi n,\ n\in\mathbb{Z}$.

10. Solve for all real $x$: $\cos(3x)=0$.

    $x=\tfrac{\pi}{2}+\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+\tfrac{\pi}{3}n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+\tfrac{\pi}{2}n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+\tfrac{\pi}{3}n,\ n\in\mathbb{Z}$.

11. Solve for all real $x$: $\sin\!\left(x-\tfrac{\pi}{4}\right)=\tfrac{1}{2}$.

    $x=\tfrac{7\pi}{12}+2\pi n$ or $x=\tfrac{11\pi}{12}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{5\pi}{12}+\pi n$ or $x=\tfrac{13\pi}{12}+\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+2\pi n$ or $x=\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{5\pi}{12}+2\pi n$ or $x=\tfrac{13\pi}{12}+2\pi n,\ n\in\mathbb{Z}$.

12. Solve for all real $x$: $\cos\!\left(x+\tfrac{\pi}{6}\right)=-\tfrac{\sqrt3}{2}$.

    $x=\pm\tfrac{5\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{5\pi}{6}+2\pi n$ or $x=\tfrac{7\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{2\pi}{3}+2\pi n$ or $x=\pi+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{2\pi}{3}+\pi n$ or $x=\pi+\pi n,\ n\in\mathbb{Z}$.

13. Solve for all real $x$: $\sin\!\left(2x-\tfrac{\pi}{3}\right)=-\tfrac{\sqrt3}{2}$.

    $x=\tfrac{4\pi}{3}+2\pi n$ or $x=\tfrac{5\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+\pi n$ or $x=\tfrac{2\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

    $x=\pi n$ or $x=\tfrac{5\pi}{6}+\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{5\pi}{6}+2\pi n$ or $x=\pi+2\pi n,\ n\in\mathbb{Z}$.

14. Solve for all real $x$: $\cos\!\left(2x+\tfrac{\pi}{4}\right)=\tfrac{\sqrt2}{2}$.

    $x=\pi n$ or $x=\tfrac{3\pi}{4}+\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{4}+2\pi n$ or $x=\tfrac{7\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\pi n$ or $x=\tfrac{3\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\pi n$ or $x=\tfrac{7\pi}{8}+\pi n,\ n\in\mathbb{Z}$.

15. Solve for all real $x$: $\sin^{2}x=\tfrac{3}{4}$.

    $x=\tfrac{\pi}{3}+2\pi n,\ \tfrac{2\pi}{3}+2\pi n,\ \tfrac{4\pi}{3}+2\pi n,\ \tfrac{5\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{2\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+2\pi n,\ \tfrac{5\pi}{6}+2\pi n,\ \tfrac{7\pi}{6}+2\pi n,\ \tfrac{11\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

16. Solve for all real $x$: $\cos^{2}x=\tfrac{1}{4}$.

    $x=\tfrac{\pi}{3}+2\pi n$ or $x=\tfrac{5\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{6}+2\pi n,\ \tfrac{5\pi}{6}+2\pi n,\ \tfrac{7\pi}{6}+2\pi n,\ \tfrac{11\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+2\pi n,\ \tfrac{5\pi}{3}+2\pi n,\ \tfrac{2\pi}{3}+2\pi n,\ \tfrac{4\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

17. Solve for all real $x$: $\sin(4x)=0$.

    $x=\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{4}n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{2}n,\ n\in\mathbb{Z}$.

18. Solve for all real $x$: $\cos(5x)=\tfrac{1}{2}$.

    $x=\pm\tfrac{\pi}{15}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{15}+\tfrac{2\pi}{5}n$ or $x=\tfrac{4\pi}{15}+\tfrac{2\pi}{5}n,\ n\in\mathbb{Z}$.

    $x=\pm\tfrac{\pi}{3}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{15}+\tfrac{2\pi}{5}n$ or $x=-\tfrac{\pi}{15}+\tfrac{2\pi}{5}n,\ n\in\mathbb{Z}$.

19. Solve for all real $x$: $\sin\!\left(3x+\tfrac{\pi}{6}\right)=\tfrac{\sqrt3}{2}$.

    $x=\tfrac{\pi}{9}+\tfrac{2\pi}{3}n$ or $x=\tfrac{5\pi}{18}+\tfrac{2\pi}{3}n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{18}+2\pi n$ or $x=\tfrac{\pi}{6}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{3}+\tfrac{2\pi}{3}n$ or $x=\tfrac{2\pi}{3}+\tfrac{2\pi}{3}n,\ n\in\mathbb{Z}$.

    $x=\tfrac{\pi}{18}+\tfrac{2\pi}{3}n$ or $x=\tfrac{\pi}{6}+\tfrac{2\pi}{3}n,\ n\in\mathbb{Z}$.

20. Solve for all real $x$: $\cos\!\left(\tfrac{x}{2}\right)=-\tfrac{\sqrt2}{2}$.

    $x=\tfrac{\pi}{2}+4\pi n$ or $x=\tfrac{7\pi}{2}+4\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{3\pi}{2}+2\pi n$ or $x=\tfrac{5\pi}{2}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{3\pi}{4}+2\pi n$ or $x=\tfrac{5\pi}{4}+2\pi n,\ n\in\mathbb{Z}$.

    $x=\tfrac{3\pi}{2}+4\pi n$ or $x=\tfrac{5\pi}{2}+4\pi n,\ n\in\mathbb{Z}$.