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06 Trigonometric Identities and Equations Online Quiz Questions

Use this free practice quiz with 20 questions to review 06 Trigonometric Identities and Equations, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

A trigonometric identity is true for every angle in the common domain of its expressions.

  1. A

    True

  2. B

    False

02
Choose one
1 point

Which equation is the reciprocal identity for secant?

  1. A

    sec⁡θ=1sin⁡θ\sec\theta=\frac{1}{\sin\theta}

  2. B

    sec⁡θ=1cos⁡θ\sec\theta=\frac{1}{\cos\theta}

  3. C

    sec⁡θ=cos⁡θsin⁡θ\sec\theta=\frac{\cos\theta}{\sin\theta}

  4. D

    sec⁡θ=sin⁡θcos⁡θ\sec\theta=\frac{\sin\theta}{\cos\theta}

03
Choose all
1 point

Select all identities that are valid for every angle in the common domain.

  1. A

    sin⁡(−θ)=−sin⁡θ\sin(-\theta)=-\sin\theta

  2. B

    cos⁡(−θ)=−cos⁡θ\cos(-\theta)=-\cos\theta

  3. C

    tan⁡(−θ)=−tan⁡θ\tan(-\theta)=-\tan\theta

  4. D

    sec⁡(−θ)=−sec⁡θ\sec(-\theta)=-\sec\theta

04
Fill in the blank
1 point

Complete the quotient identities: tan⁡θ=\tan\theta= and cot⁡θ=\cot\theta=.

05
Written response
1 point

What is the period of tan⁡x\tan x, in exact radians?

06
True or false
1 point

The cosine sum formula is cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta.

  1. A

    True

  2. B

    False

07
Choose one
1 point

What is the exact value of cos⁡(π12)\cos\left(\frac{\pi}{12}\right)?

  1. A

    6−24\frac{\sqrt{6}-\sqrt{2}}{4}

  2. B

    3+12\frac{\sqrt{3}+1}{2}

  3. C

    2+34\frac{\sqrt{2}+\sqrt{3}}{4}

  4. D

    2+64\frac{\sqrt{2}+\sqrt{6}}{4}

08
Fill in the blank
1 point

Complete these double-angle formulas: sin⁡(2θ)=\sin(2\theta)= and cos⁡(2θ)=\cos(2\theta)=.

09
Choose all
1 point

Select all statements that correctly describe half-angle formulas.

  1. A

    sin⁡α2=±1−cos⁡α2\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}

  2. B

    The sign in a half-angle formula is determined by the quadrant containing α\alpha.

  3. C

    tan⁡α2=sin⁡α1+cos⁡α\tan\frac{\alpha}{2}=\frac{\sin\alpha}{1+\cos\alpha}, when defined

  4. D

    tan⁡α2=1−cos⁡αsin⁡α\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}, when defined

10
Written response
1 point

How many solutions does tan⁡x=1\tan x=1 have on the interval [0,2π)[0,2\pi)?

11
Choose one
1 point

Solve 2cos⁡x−1=02\cos x-1=0 on [0,2π)[0,2\pi).

  1. A

    x=π3,5π3x=\frac{\pi}{3},\frac{5\pi}{3}

  2. B

    x=π3,2π3x=\frac{\pi}{3},\frac{2\pi}{3}

  3. C

    x=2π3,4π3x=\frac{2\pi}{3},\frac{4\pi}{3}

  4. D

    x=5π6,7π6x=\frac{5\pi}{6},\frac{7\pi}{6}

12
Open ended
1 point

Simplify 1+tan⁡2θsec⁡θ\frac{1+\tan^2\theta}{\sec\theta} and state the domain qualification needed for the simplification.

13
Choose one
1 point

Which expression equals tan⁡(α−β)\tan(\alpha-\beta), wherever the formula is defined?

  1. A

    tan⁡α+tan⁡β1+tan⁡αtan⁡β\frac{\tan\alpha+\tan\beta}{1+\tan\alpha\tan\beta}

  2. B

    tan⁡α−tan⁡β1+tan⁡αtan⁡β\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}

  3. C

    tan⁡α−tan⁡β1−anαtan⁡β\frac{\tan\alpha-\tan\beta}{1- an\alpha\tan\beta}

  4. D

    tan⁡α+tan⁡β1−anαtan⁡β\frac{\tan\alpha+\tan\beta}{1- an\alpha\tan\beta}

14
True or false
1 point

True or false: A trigonometric identity must be true for every angle in the common domain of the expressions, whereas a trigonometric equation is generally true only for particular values of the variable.

  1. A

    True

  2. B

    False

15
Choose one
1 point

Which expression is equivalent to cos⁡(α−β)\cos(\alpha-\beta)?

  1. A

    cos⁡αcos⁡β−sin⁡αsin⁡β\cos\alpha\cos\beta-\sin\alpha\sin\beta

  2. B

    sin⁡αcos⁡β−cos⁡αsin⁡β\sin\alpha\cos\beta-\cos\alpha\sin\beta

  3. C

    cos⁡αcos⁡β+sin⁡αsin⁡β\cos\alpha\cos\beta+\sin\alpha\sin\beta

  4. D

    sin⁡αsin⁡β−cos⁡αcos⁡β\sin\alpha\sin\beta-\cos\alpha\cos\beta

16
Written response
1 point

How many solutions does 2cos⁡x=12\cos x=1 have on the interval [0,2π)[0,2\pi)? Enter the exact number of solutions.

17
Choose one
1 point

Solve 2sin⁡2x−sin⁡x−1=02\sin^2x-\sin x-1=0 on [0,2π)[0,2\pi). Which set contains all solutions?

  1. A

    π6,5π6,3π2\frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2}

  2. B

    π2,7π6,11π6\frac{\pi}{2},\frac{7\pi}{6},\frac{11\pi}{6}

  3. C

    π2,5π6,7π6\frac{\pi}{2},\frac{5\pi}{6},\frac{7\pi}{6}

  4. D

    π6,3π2,11π6\frac{\pi}{6},\frac{3\pi}{2},\frac{11\pi}{6}

18
Choose one
1 point

What is the exact value of sin⁡(π8)\sin\left(\frac{\pi}{8}\right)?

  1. A

    2−22\frac{\sqrt{2-\sqrt{2}}}{2}

  2. B

    −2−22-\frac{\sqrt{2-\sqrt{2}}}{2}

  3. C

    2+22\frac{\sqrt{2+\sqrt{2}}}{2}

  4. D

    −2+22-\frac{\sqrt{2+\sqrt{2}}}{2}

19
Choose one
1 point

Solve cos⁡xcos⁡(2x)+sin⁡xsin⁡(2x)=32\cos x\cos(2x)+\sin x\sin(2x)=\frac{\sqrt{3}}{2} on [0,2π)[0,2\pi).

  1. A

    π3,5π3\frac{\pi}{3},\frac{5\pi}{3}

  2. B

    π6,11π6\frac{\pi}{6},\frac{11\pi}{6}

  3. C

    2π3,4π3\frac{2\pi}{3},\frac{4\pi}{3}

  4. D

    5π6,7π6\frac{5\pi}{6},\frac{7\pi}{6}

20
Written response
1 point

Using the sine difference formula, find the exact value of sin⁡(π12)\sin\left(\frac{\pi}{12}\right).