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← Trigonometry
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Practice deliberately

Work through Trigonometry unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 6 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Trigonometry coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Trigonometry syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use precalculus practice questions with answers.

01

Angle measure

Connect angles, coordinates, ratios, and graphs while keeping degree or radian measure explicit and checking all solutions in the interval.

Goal: recognise a angle measure problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Unit circle

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a unit circle problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Graphs of sine and cosine

Connect angles, coordinates, ratios, and graphs while keeping degree or radian measure explicit and checking all solutions in the interval.

Goal: recognise a graphs of sine and cosine problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Identities

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a identities problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Trig equations

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a trig equations problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Sine and cosine laws

Connect angles, coordinates, ratios, and graphs while keeping degree or radian measure explicit and checking all solutions in the interval.

Goal: recognise a sine and cosine laws problem from its wording, carry out the governing method, and check that the result is reasonable.

Trigonometry questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 6 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Convert 210 degrees to radians.

Trigonometry · Angle measure

  1. 7 pi / 6
  2. 5 pi / 6
  3. 7 pi / 4
  4. 11 pi / 6
Show the worked answer
  1. Multiply degrees by pi / 180 to get radians.
  2. 210 x pi / 180 = 210 pi / 180.
  3. Divide top and bottom by 30: 210 / 30 = 7 and 180 / 30 = 6.
  4. So 210 degrees is 7 pi / 6 radians.

Answer: 7 pi / 6

Question 2: A circle has a radius of 8 cm. What arc length does a central angle of 2.5 radians cut off?

Trigonometry · Angle measure

  1. 20 cm
  2. 3.2 cm
  3. 12.6 cm
  4. 50 cm
Show the worked answer
  1. Arc length equals radius times the angle, when the angle is measured in radians.
  2. There is no need to convert, because 2.5 is already in radians.
  3. 8 x 2.5 = 20.
  4. The arc is 20 cm long.

Answer: 20 cm

Question 3: What is the exact value of sin(5 pi / 3)?

Trigonometry · Unit circle

  1. negative square root of 3, over 2
  2. -1/2
  3. square root of 3, over 2
  4. 1/2
Show the worked answer
  1. 5 pi / 3 is pi / 3 short of a full turn, so it sits in the fourth quadrant.
  2. Sine is negative in the fourth quadrant.
  3. The reference angle is pi / 3, and sin(pi / 3) = square root of 3, over 2.
  4. So sin(5 pi / 3) = negative square root of 3, over 2.

Answer: negative square root of 3, over 2

Question 4: For y = 3 sin(2x - pi), what are the amplitude and the period?

Trigonometry · Graphs of sine and cosine

  1. amplitude 3, period pi
  2. amplitude 3, period 2 pi
  3. amplitude 6, period pi
  4. amplitude 2, period 3
Show the worked answer
  1. The number in front of sine sets the amplitude, so the amplitude is 3.
  2. The period of sine is 2 pi divided by the number multiplying x.
  3. Here that number is 2.
  4. 2 pi / 2 = pi, so the period is pi.

Answer: amplitude 3, period pi

Question 5: If sin(theta) = 3/5 and theta is in the second quadrant, what is tan(theta)?

Trigonometry · Identities

  1. -3/4
  2. 3/4
  3. -4/3
  4. 4/5
Show the worked answer
  1. Use sin^2 + cos^2 = 1, so cos^2 = 1 - (3/5)^2 = 1 - 9/25 = 16/25.
  2. That gives cos = 4/5 or -4/5.
  3. In the second quadrant cosine is negative, so cos(theta) = -4/5.
  4. tan = sin / cos = (3/5) / (-4/5) = -3/4.

Answer: -3/4

Question 6: Solve 2 cos(x) + 1 = 0 for x between 0 and 2 pi.

Trigonometry · Trig equations

  1. x = 2 pi / 3 and x = 4 pi / 3
  2. x = pi / 3 and x = 5 pi / 3
  3. x = pi / 6 and x = 11 pi / 6
  4. x = pi / 2 and x = 3 pi / 2
Show the worked answer
  1. Rearrange to cos(x) = -1/2.
  2. The reference angle with cosine 1/2 is pi / 3.
  3. Cosine is negative in the second and third quadrants.
  4. Second quadrant: pi - pi/3 = 2 pi / 3. Third quadrant: pi + pi/3 = 4 pi / 3.

Answer: x = 2 pi / 3 and x = 4 pi / 3

Question 7: In a triangle, angle A = 40 degrees, angle B = 75 degrees, and side a = 12. Find side b to one decimal place.

Trigonometry · Sine and cosine laws

  1. 18.0
  2. 7.9
  3. 13.5
  4. 20.4
Show the worked answer
  1. The law of sines says a / sin A = b / sin B.
  2. Rearranged, b = a x sin B / sin A = 12 x sin 75 / sin 40.
  3. sin 75 is about 0.9659 and sin 40 is about 0.6428.
  4. 12 x 0.9659 / 0.6428 = 18.0 to one decimal place.

Answer: 18.0

Question 8: Two sides of a triangle are 7 and 10, and the angle between them is 60 degrees. How long is the third side, to two decimal places?

Trigonometry · Sine and cosine laws

  1. 8.89
  2. 12.21
  3. 6.24
  4. 9.43
Show the worked answer
  1. Use the law of cosines: c^2 = a^2 + b^2 - 2ab cos(C).
  2. 7^2 + 10^2 = 49 + 100 = 149.
  3. 2 x 7 x 10 x cos 60 = 140 x 0.5 = 70.
  4. c^2 = 149 - 70 = 79, and the square root of 79 is 8.89.

Answer: 8.89

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Trigonometry questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Angle measureConvert 210 degrees to radians.7 pi / 6
2Angle measureA circle has a radius of 8 cm. What arc length does a central angle of 2.5 radians cut off?20 cm
3Unit circleWhat is the exact value of sin(5 pi / 3)?negative square root of 3, over 2
4Graphs of sine and cosineFor y = 3 sin(2x - pi), what are the amplitude and the period?amplitude 3, period pi
5IdentitiesIf sin(theta) = 3/5 and theta is in the second quadrant, what is tan(theta)?-3/4
6Trig equationsSolve 2 cos(x) + 1 = 0 for x between 0 and 2 pi.x = 2 pi / 3 and x = 4 pi / 3
7Sine and cosine lawsIn a triangle, angle A = 40 degrees, angle B = 75 degrees, and side a = 12. Find side b to one decimal place.18.0
8Sine and cosine lawsTwo sides of a triangle are 7 and 10, and the angle between them is 60 degrees. How long is the third side, to two decimal places?8.89

How the generated Trigonometry sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Trigonometry practice page

What does this page cover?

It covers all 6 Trigonometry units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.