Chapter Check · Full validated bank

Chapter Check: Trigonometric Functions

Every reviewed question in this chapter's tagged bank, shuffled. No hints — submit once to see your score by concept band and difficulty.

A Angles and Radian Measure · 3B Unit Circle, Signs and Standard Values · 8C Domains, Ranges and Periodicity · 6D Compound Angles · 4E Multiple-Angle and Transformation Identities · 3F Trigonometric Equations · 6Total 30 questions

FoundationalBand 2 · Unit Circle

1. From the displayed unit circle, the sign of $\cos120^\circ$ is

positive negative zero undefined

ApplicationBand 3 · Periodicity

2. $\cos765^\circ$ equals

$-1/\sqrt2$ $1$ $0$ $1/\sqrt2$

ApplicationBand 3 · Periodicity

3. $\sin390^\circ$ equals

$\sqrt3/2$ $-1/2$ $-\sqrt3/2$ $1/2$

ChallengingBand 6 · General Solutions

4. The general solution of $\sin x=\sin\alpha$ is

$x=2n\pi\pm\alpha$ $x=n\pi+(-1)^n\alpha$ $x=2n\pi+\alpha$ $x=n\pi+\alpha$

FoundationalBand 1 · Degree–Radian Conversion

5. Convert $225^\circ$ to radians.

$5\pi/4$ $3\pi/4$ $7\pi/6$ $4\pi/5$

ApplicationBand 5 · Double Angle

6. If $\tan x=1/2$, then $\tan2x$ is

$4/3$ $1$ $3/4$ $2$

ApplicationBand 2 · Unit Circle

7. The point on the unit circle at $150^\circ$ has coordinates

$(1/2,-\sqrt3/2)$ $(\sqrt3/2,1/2)$ $(-\sqrt3/2,1/2)$ $(-1/2,\sqrt3/2)$

ApplicationBand 5 · Transformations

8. $2\sin A\cos B$ equals

$\sin(A+B)-\sin(A-B)$ $\cos(A-B)-\cos(A+B)$ $2\cos A\sin B$ $\sin(A+B)+\sin(A-B)$

FoundationalBand 2 · Pythagorean Identity

9. The value of $\cosec^2x-\cot^2x$ is

$1$ $-1$ $2$ $0$

FoundationalBand 3 · Periodicity

10. The fundamental period of $\tan x$ is

$\pi/2$ $\pi$ $2\pi$ $4\pi$

ApplicationBand 4 · Compound Angle

11. $\sin15^\circ$ equals

$(\sqrt6-\sqrt2)/4$ $(\sqrt6+\sqrt2)/4$ $\sqrt2/2$ $1/2$

FoundationalBand 2 · Odd–Even Properties

12. $\sin(-x)$ equals

$-\sin x$ $\cos x$ $-\cos x$ $\sin x$

ApplicationBand 5 · Double Angle

13. If $\sin x=5/13$ and $\cos x=12/13$, then $\cos2x$ is

$144/169$ $119/169$ $-119/169$ $120/169$

ApplicationBand 3 · Domain and Range

14. The domain of $\tan x$ excludes

$2n\pi$ $n\pi/4$ $n\pi$ $(2n+1)\pi/2$

ApplicationBand 6 · Principal Solutions

15. In $0\le x<2\pi$, $\cos x=-1/2$ has solutions

$\pi/6,5\pi/6$ $3\pi/4,5\pi/4$ $2\pi/3,4\pi/3$ $\pi/3,5\pi/3$

FoundationalBand 6 · General Solutions

16. The general solution of $\cos x=1$ is

$x=n\pi$ $x=(2n+1)\pi$ $x=n\pi+\pi/2$ $x=2n\pi$

FoundationalBand 2 · Odd–Even Properties

17. $\cos(-x)$ equals

$\sin x$ $-\sin x$ $\cos x$ $-\cos x$

ApplicationBand 6 · Principal Solutions

18. In $0\le x<2\pi$, $\tan x=-1$ has solutions

$5\pi/4,7\pi/4$ $3\pi/4,7\pi/4$ $\pi/4,5\pi/4$ $\pi/4,3\pi/4$

ApplicationBand 4 · Compound Angle

19. $\cos15^\circ$ equals

$1/2$ $(\sqrt6+\sqrt2)/4$ $(\sqrt6-\sqrt2)/4$ $\sqrt3/2$

FoundationalBand 1 · Degree–Radian Conversion

20. Convert $7\pi/6$ radians to degrees.

$210^\circ$ $240^\circ$ $150^\circ$ $180^\circ$

FoundationalBand 6 · General Solutions

21. The general solution of $\sin x=0$ is

$x=n\pi$ $x=2n\pi$ $x=(2n+1)\pi/2$ $x=n\pi+\pi/4$

ApplicationBand 4 · Compound Angle

22. If $\sin A=3/5,\cos A=4/5,\sin B=5/13,\cos B=12/13$, then $\sin(A+B)$ is

$63/65$ $33/65$ $16/65$ $56/65$

FoundationalBand 3 · Periodicity

23. The fundamental period of $\cos x$ is

$4\pi$ $\pi/2$ $\pi$ $2\pi$

FoundationalBand 6 · General Solutions

24. The general solution of $\cos x=0$ is

$x=2n\pi$ $x=n\pi+\pi/4$ $x=(2n+1)\pi/2$ $x=n\pi$

ApplicationBand 3 · Graph of Sine

25. The displayed graph has range

$\mathbb R$ $[0,1]$ $(-\infty,1]$ $[-1,1]$

FoundationalBand 2 · Quadrant Signs

26. In Quadrant III, which trigonometric function is positive?

$\sin x$ $\cos x$ $\tan x$ $\cosec x$

FoundationalBand 2 · Pythagorean Identity

27. The value of $\sec^2x-\tan^2x$ is

$0$ $1$ $-1$ $2$

FoundationalBand 1 · Arc Length

28. For $r=12$ cm and $\theta=\pi/4$, the arc length is

$4\pi$ $6\pi$ $2\pi$ $3\pi$

FoundationalBand 2 · Pythagorean Identity

29. If $\cos x=12/13$ and $x$ is acute, then $\sin x$ is

$-5/13$ $5/13$ $12/5$ $13/5$

ApplicationBand 4 · Tangent Difference

30. If $\tan A=2$ and $\tan B=1/3$, then $\tan(A-B)$ is

$7/5$ $5/7$ $1$ $5$