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Exact trigonometric values practice problems

Exact values of the six trig functions at multiples of π/6 and π/4, and degrees to radians. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 13.46

easy
Find the exact value of \( \displaystyle \sin{\left(- \frac{\pi}{3} \right)} \).
  1. The angle -pi/3 ends in quadrant 4 with reference angle pi/3, where sin is negative.
    Reviewed
  2. \[ \sin{\left(\frac{\pi}{3} \right)} = \frac{\sqrt{3}}{2} \]
    sin of the reference angle.✓ Proved
  3. \[ \sin{\left(\frac{\left(-1\right) \pi}{3} \right)} = - \frac{\sqrt{3}}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \frac{\sqrt{3}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.47

easy
Find the exact value of \( \displaystyle \cos{\left(\frac{\pi}{6} \right)} \).
  1. The angle pi/6 ends in quadrant 1 with reference angle pi/6, where cos is positive.
    Reviewed
  2. \[ \cos{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{2} \]
    cos of the reference angle.✓ Proved
  3. \[ \cos{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \frac{\sqrt{3}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.48

easy
Find the exact value of \( \displaystyle \csc{\left(\frac{\pi}{3} \right)} \).
  1. The angle pi/3 ends in quadrant 1 with reference angle pi/3, where csc is positive.
    Reviewed
  2. \[ \csc{\left(\frac{\pi}{3} \right)} = \frac{2 \sqrt{3}}{3} \]
    csc of the reference angle.✓ Proved
  3. \[ \csc{\left(\frac{\pi}{3} \right)} = \frac{2 \sqrt{3}}{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \frac{2 \sqrt{3}}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.50

easy
Find the exact value of \( \displaystyle \tan{\left(\frac{5 \pi}{6} \right)} \).
  1. The angle 5*pi/6 ends in quadrant 2 with reference angle pi/6, where tan is negative.
    Reviewed
  2. \[ \tan{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{3} \]
    tan of the reference angle.✓ Proved
  3. \[ \tan{\left(\frac{5 \pi}{6} \right)} = - \frac{\sqrt{3}}{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \frac{\sqrt{3}}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.51

easy
Find the exact value of \( \displaystyle \csc{\left(- \frac{7 \pi}{6} \right)} \).
  1. The angle -7*pi/6 ends in quadrant 2 with reference angle pi/6, where csc is positive.
    Reviewed
  2. \[ \csc{\left(\frac{\pi}{6} \right)} = 2 \]
    csc of the reference angle.✓ Proved
  3. \[ \csc{\left(- \frac{7 \pi}{6} \right)} = 2 \]
    Attach the sign for the quadrant.✓ Proved
Answer \( 2 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.53

easy
Find the exact value of \( \displaystyle \cot{\left(\frac{7 \pi}{6} \right)} \).
  1. The angle 7*pi/6 ends in quadrant 3 with reference angle pi/6, where cot is positive.
    Reviewed
  2. \[ \cot{\left(\frac{\pi}{6} \right)} = \sqrt{3} \]
    cot of the reference angle.✓ Proved
  3. \[ \cot{\left(\frac{7 \pi}{6} \right)} = \sqrt{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \sqrt{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.55

easy
Find the exact value of \( \displaystyle \sec{\left(\frac{\pi}{3} \right)} \).
  1. The angle pi/3 ends in quadrant 1 with reference angle pi/3, where sec is positive.
    Reviewed
  2. \[ \sec{\left(\frac{\pi}{3} \right)} = 2 \]
    sec of the reference angle.✓ Proved
  3. \[ \sec{\left(\frac{\pi}{3} \right)} = 2 \]
    Attach the sign for the quadrant.✓ Proved
Answer \( 2 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.58

easy
Find the exact value of \( \displaystyle \sin{\left(\frac{2 \pi}{3} \right)} \).
  1. The angle 2*pi/3 ends in quadrant 2 with reference angle pi/3, where sin is positive.
    Reviewed
  2. \[ \sin{\left(\frac{\pi}{3} \right)} = \frac{\sqrt{3}}{2} \]
    sin of the reference angle.✓ Proved
  3. \[ \sin{\left(\frac{2 \pi}{3} \right)} = \frac{\sqrt{3}}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \frac{\sqrt{3}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.59

easy
Find the exact value of \( \displaystyle \csc{\left(\frac{11 \pi}{6} \right)} \).
  1. The angle 11*pi/6 ends in quadrant 4 with reference angle pi/6, where csc is negative.
    Reviewed
  2. \[ \csc{\left(\frac{\pi}{6} \right)} = 2 \]
    csc of the reference angle.✓ Proved
  3. \[ \csc{\left(\frac{11 \pi}{6} \right)} = -2 \]
    Attach the sign for the quadrant.✓ Proved
Answer \( -2 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.236

easy
Find the exact value of \( \displaystyle \sec{\left(- \frac{3 \pi}{4} \right)} \).
  1. The angle -3*pi/4 ends in quadrant 3 with reference angle pi/4, where sec is negative.
    Reviewed
  2. \[ \sec{\left(\frac{\pi}{4} \right)} = \sqrt{2} \]
    sec of the reference angle.✓ Proved
  3. \[ \sec{\left(- \frac{3 \pi}{4} \right)} = - \sqrt{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \sqrt{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.238

easy
Find the exact value of \( \displaystyle \cos{\left(- \frac{3 \pi}{4} \right)} \).
  1. The angle -3*pi/4 ends in quadrant 3 with reference angle pi/4, where cos is negative.
    Reviewed
  2. \[ \cos{\left(\frac{\pi}{4} \right)} = \frac{\sqrt{2}}{2} \]
    cos of the reference angle.✓ Proved
  3. \[ \cos{\left(- \frac{3 \pi}{4} \right)} = - \frac{\sqrt{2}}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \frac{\sqrt{2}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.240

easy
Find the exact value of \( \displaystyle \tan{\left(- \frac{5 \pi}{3} \right)} \).
  1. The angle -5*pi/3 ends in quadrant 1 with reference angle pi/3, where tan is positive.
    Reviewed
  2. \[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]
    tan of the reference angle.✓ Proved
  3. \[ \tan{\left(- \frac{5 \pi}{3} \right)} = \sqrt{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \sqrt{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.242

easy
Find the exact value of \( \displaystyle \cot{\left(- \frac{11 \pi}{6} \right)} \).
  1. The angle -11*pi/6 ends in quadrant 1 with reference angle pi/6, where cot is positive.
    Reviewed
  2. \[ \cot{\left(\frac{\pi}{6} \right)} = \sqrt{3} \]
    cot of the reference angle.✓ Proved
  3. \[ \cot{\left(- \frac{11 \pi}{6} \right)} = \sqrt{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \sqrt{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.243

easy
Find the exact value of \( \displaystyle \tan{\left(- \frac{\pi}{3} \right)} \).
  1. The angle -pi/3 ends in quadrant 4 with reference angle pi/3, where tan is negative.
    Reviewed
  2. \[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]
    tan of the reference angle.✓ Proved
  3. \[ \tan{\left(\frac{\left(-1\right) \pi}{3} \right)} = - \sqrt{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \sqrt{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.244

easy
Find the exact value of \( \displaystyle \sin{\left(\frac{\pi}{6} \right)} \).
  1. The angle pi/6 ends in quadrant 1 with reference angle pi/6, where sin is positive.
    Reviewed
  2. \[ \sin{\left(\frac{\pi}{6} \right)} = \frac{1}{2} \]
    sin of the reference angle.✓ Proved
  3. \[ \sin{\left(\frac{\pi}{6} \right)} = \frac{1}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \frac{1}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.249

easy
Find the exact value of \( \displaystyle \sec{\left(\frac{5 \pi}{6} \right)} \).
  1. The angle 5*pi/6 ends in quadrant 2 with reference angle pi/6, where sec is negative.
    Reviewed
  2. \[ \sec{\left(\frac{\pi}{6} \right)} = \frac{2 \sqrt{3}}{3} \]
    sec of the reference angle.✓ Proved
  3. \[ \sec{\left(\frac{5 \pi}{6} \right)} = - \frac{2 \sqrt{3}}{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \frac{2 \sqrt{3}}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.336

easy
Find the exact value of \( \displaystyle \csc{\left(- \frac{\pi}{4} \right)} \).
  1. The angle -pi/4 ends in quadrant 4 with reference angle pi/4, where csc is negative.
    Reviewed
  2. \[ \csc{\left(\frac{\pi}{4} \right)} = \sqrt{2} \]
    csc of the reference angle.✓ Proved
  3. \[ \csc{\left(\frac{\left(-1\right) \pi}{4} \right)} = - \sqrt{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \sqrt{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.340

easy
Find the exact value of \( \displaystyle \sin{\left(- \frac{7 \pi}{4} \right)} \).
  1. The angle -7*pi/4 ends in quadrant 1 with reference angle pi/4, where sin is positive.
    Reviewed
  2. \[ \sin{\left(\frac{\pi}{4} \right)} = \frac{\sqrt{2}}{2} \]
    sin of the reference angle.✓ Proved
  3. \[ \sin{\left(- \frac{7 \pi}{4} \right)} = \frac{\sqrt{2}}{2} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \frac{\sqrt{2}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.341

easy
Find the exact value of \( \displaystyle \cot{\left(\frac{\pi}{6} \right)} \).
  1. The angle pi/6 ends in quadrant 1 with reference angle pi/6, where cot is positive.
    Reviewed
  2. \[ \cot{\left(\frac{\pi}{6} \right)} = \sqrt{3} \]
    cot of the reference angle.✓ Proved
  3. \[ \cot{\left(\frac{\pi}{6} \right)} = \sqrt{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( \sqrt{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.411

easy
Find the exact value of \( \displaystyle \cot{\left(- \frac{\pi}{3} \right)} \).
  1. The angle -pi/3 ends in quadrant 4 with reference angle pi/3, where cot is negative.
    Reviewed
  2. \[ \cot{\left(\frac{\pi}{3} \right)} = \frac{\sqrt{3}}{3} \]
    cot of the reference angle.✓ Proved
  3. \[ \cot{\left(\frac{\left(-1\right) \pi}{3} \right)} = - \frac{\sqrt{3}}{3} \]
    Attach the sign for the quadrant.✓ Proved
Answer \( - \frac{\sqrt{3}}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More exact trigonometric values practice

Convert \( \displaystyle -45^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.49easy✓ Nihil obstat
Convert \( \displaystyle 225^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.52easy✓ Nihil obstat
Convert \( \displaystyle 30^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.54easy✓ Nihil obstat
Convert \( \displaystyle 150^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.56easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sin{\left(- \frac{\pi}{4} \right)} \).
Problem 13.57easy✓ Nihil obstat
Convert \( \displaystyle 15^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.60easy✓ Nihil obstat
Convert \( \displaystyle 240^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.235easy✓ Nihil obstat
Convert \( \displaystyle -120^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.237easy✓ Nihil obstat
Convert \( \displaystyle 120^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.239easy✓ Nihil obstat
Find the exact value of \( \displaystyle \csc{\left(\frac{\pi}{4} \right)} \).
Problem 13.241easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(- \frac{4 \pi}{3} \right)} \).
Problem 13.245easy✓ Every equation proved
Convert \( \displaystyle 210^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.246easy✓ Nihil obstat
Convert \( \displaystyle 400^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.247easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(\frac{4 \pi}{3} \right)} \).
Problem 13.248easy✓ Every equation proved
Convert \( \displaystyle 300^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.332easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sin{\left(\frac{5 \pi}{4} \right)} \).
Problem 13.333easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cot{\left(\frac{11 \pi}{6} \right)} \).
Problem 13.334easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cot{\left(\frac{4 \pi}{3} \right)} \).
Problem 13.335easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sin{\left(\frac{3 \pi}{4} \right)} \).
Problem 13.337easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sin{\left(\frac{5 \pi}{6} \right)} \).
Problem 13.338easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sec{\left(\frac{\pi}{4} \right)} \).
Problem 13.339easy✓ Nihil obstat
Find the exact value of \( \displaystyle \csc{\left(- \frac{5 \pi}{6} \right)} \).
Problem 13.412easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sin{\left(\frac{7 \pi}{4} \right)} \).
Problem 13.413easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cos{\left(\frac{5 \pi}{6} \right)} \).
Problem 13.414easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cos{\left(\frac{5 \pi}{3} \right)} \).
Problem 13.415easy✓ Nihil obstat
Convert \( \displaystyle 270^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.416easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(\frac{7 \pi}{6} \right)} \).
Problem 13.417easy✓ Every equation proved
Find the exact value of \( \displaystyle \tan{\left(\frac{3 \pi}{4} \right)} \).
Problem 13.418easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(\frac{7 \pi}{4} \right)} \).
Problem 13.419easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cos{\left(\frac{11 \pi}{6} \right)} \).
Problem 13.420easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cot{\left(\frac{\pi}{4} \right)} \).
Problem 13.489easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(\frac{\pi}{3} \right)} \).
Problem 13.490easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sec{\left(- \frac{11 \pi}{6} \right)} \).
Problem 13.491easy✓ Nihil obstat
Convert \( \displaystyle 330^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.492easy✓ Nihil obstat
Find the exact value of \( \displaystyle \csc{\left(- \frac{7 \pi}{4} \right)} \).
Problem 13.493easy✓ Nihil obstat
Find the exact value of \( \displaystyle \tan{\left(\frac{5 \pi}{4} \right)} \).
Problem 13.494easy✓ Nihil obstat
Convert \( \displaystyle 135^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
Problem 13.495easy✓ Nihil obstat
Find the exact value of \( \displaystyle \sec{\left(\frac{\pi}{6} \right)} \).
Problem 13.496easy✓ Nihil obstat
Find the exact value of \( \displaystyle \csc{\left(\frac{5 \pi}{3} \right)} \).
Problem 13.497easy✓ Nihil obstat
Find the exact value of \( \displaystyle \cos{\left(\frac{5 \pi}{4} \right)} \).
Problem 13.498easy✓ Nihil obstat