Trigonometry Worksheets

Find exact values of sine, cosine and tangent at special angles and solve basic trigonometric equations. Grades 11–12.

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What these worksheets practise

A handful of angles (0°, 30°, 45°, 60°, 90° and their matches in the other quadrants) have sine, cosine and tangent values that can be written exactly, using fractions and square roots. They come from two special right triangles: the 45°-45°-90° triangle with sides 1,1,21, 1, \sqrt{2}, and the 30°-60°-90° triangle with sides 1,3,21, \sqrt{3}, 2.

Beyond 90°, every angle has a reference angle (its acute angle to the xx-axis). The value has the same size as at the reference angle, and the quadrant decides the sign.

  • Easy — exact values at 0°, 30°, 45°, 60° and 90° (first quadrant).
  • Medium — exact values at special angles anywhere from 0° to 330°.
  • Hard — solve sin⁡(x)=k\sin(x) = k or cos⁡(x)=k\cos(x) = k for both solutions in [0°,360°)[0°, 360°).

Worked examples

Easy: cos⁡(30°)\cos(30°)

In the 30°-60°-90° triangle, the side next to the 30° angle is 3\sqrt{3} and the hypotenuse is 2.

cos⁡(30°)=adjacenthypotenuse=32\cos(30°) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}

Medium: sin⁡(240°)\sin(240°)

240° is in the third quadrant, and its reference angle is 240°−180°=60°240° - 180° = 60°. Sine is negative in the third quadrant.

sin⁡(240°)=−sin⁡(60°)=−32\sin(240°) = -\sin(60°) = -\frac{\sqrt{3}}{2}

Hard: cos⁡(x)=−12\cos(x) = -\tfrac{1}{2} for 0°≤x<360°0° \le x < 360°

cos⁡(60°)=12\cos(60°) = \tfrac{1}{2}, so the reference angle is 60°. Cosine is negative in the second and third quadrants:

x=180°−60°=120°x=180°+60°=240°x = 180° - 60° = 120° \qquad x = 180° + 60° = 240°

Answer: x=120°, 240°x = 120°,\ 240°.

Check your answers

A calculator in degree mode gives cos⁡(120°)=−0.5\cos(120°) = -0.5 and cos⁡(240°)=−0.5\cos(240°) = -0.5, and sin⁡(240°)≈−0.866\sin(240°) \approx -0.866, which matches −32-\tfrac{\sqrt{3}}{2}.

Common mistakes

  • Wrong sign for the quadrant. Sine is positive in quadrants I and II, cosine in I and IV, tangent in I and III. sin⁡(240°)\sin(240°) is negative.
  • Giving only one solution. Between 0° and 360°, cos⁡(x)=−12\cos(x) = -\tfrac{1}{2} has two solutions. Stopping at 120° misses 240°.
  • Mixing up sine and cosine at 30° and 60°. sin⁡(30°)=12\sin(30°) = \tfrac{1}{2} but cos⁡(30°)=32\cos(30°) = \tfrac{\sqrt{3}}{2}. Drawing the triangle settles it.
  • Calculator in radian mode. Checking cos⁡(120)\cos(120) in radians gives about 0.814, which looks like a wrong answer but is only the wrong mode.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. tan⁡(45°)=\tan(45°) =
  2. sin⁡(45°)=\sin(45°) =
  3. sin⁡(90°)=\sin(90°) =

Medium

  1. sin⁡(90°)=\sin(90°) =
  2. cos⁡(150°)=\cos(150°) =
  3. cos⁡(180°)=\cos(180°) =

Hard

  1. cos⁡(x)=22\cos(x) = \frac{\sqrt{2}}{2}
    Find x ∈ [0°, 360°)
  2. sin⁡(x)=12\sin(x) = \frac{1}{2}
    Find x ∈ [0°, 360°)
  3. sin⁡(x)=32\sin(x) = \frac{\sqrt{3}}{2}
    Find x ∈ [0°, 360°)
Show answers
  1. 1
  2. √2/2
  3. 1
  4. 1
  5. −√3/2
  6. −1
  7. x = 45°, 315°
  8. x = 30°, 150°
  9. x = 60°, 120°

Tips for teachers and parents

  • Have students sketch the two special triangles at the top of every Easy worksheet until they can recall the values without them.
  • For Medium and Hard, a quick sketch of the angle on a set of axes makes the reference angle and the sign visible.
  • tan⁡(90°)\tan(90°) and tan⁡(270°)\tan(270°) are undefined, so they never appear as problems; they make a good discussion question.

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