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Examples · Trigonometry

Trig Examples: 8 Worked Problems

Eight angles, ordered so each one adds exactly one new idea to the last. Every value is checked against the sign of its quadrant.

7 min read Updated 2026-10-05 Checked by Aziza Smailovic

The second quadrant

Angles between 90° and 180° have a reference angle of 180° minus the angle, and only sine is positive.

1. sin 120°worked example
  1. 120°between 90° and 180°, so quadrant II
  2. 180° − 120° = 60°the reference angle
  3. sin 60° = √3/2the size of the answer
  4. sin 120° = √3/2sine is positive in quadrant II
2. tan 135°worked example
  1. 135°quadrant II
  2. 180° − 135° = 45°the reference angle
  3. tan 45° = 1the size of the answer
  4. tan 135° = −1tangent is negative in quadrant II

The third quadrant

Angles between 180° and 270° have a reference angle of the angle minus 180°, and only tangent is positive.

3. cos 225°worked example
  1. 225°between 180° and 270°, so quadrant III
  2. 225° − 180° = 45°the reference angle
  3. cos 45° = √2/2the size of the answer
  4. cos 225° = −√2/2cosine is negative in quadrant III
4. csc 210°worked example
  1. 210°quadrant III
  2. 210° − 180° = 30°the reference angle
  3. csc 30° = 2the size of the answer: csc is 1 ÷ sin, and sin 30° = 1/2
  4. csc 210° = −2sine is negative in quadrant III, so its reciprocal is too

The fourth quadrant

Angles between 270° and 360° have a reference angle of 360° minus the angle, and only cosine is positive.

5. tan 300°worked example
  1. 300°between 270° and 360°, so quadrant IV
  2. 360° − 300° = 60°the reference angle
  3. tan 60° = √3the size of the answer
  4. tan 300° = −√3tangent is negative in quadrant IV

A large angle and a negative angle

6. sin 510°worked example
  1. 510° − 360° = 150°remove one whole turn to find the coterminal angle
  2. 150°quadrant II, reference angle 180° − 150° = 30°
  3. sin 150° = 1/2sine is positive in quadrant II
7. cos (−60°)worked example
  1. −60° + 360° = 300°add a turn to bring the angle into 0° to 360°
  2. 300°quadrant IV, reference angle 360° − 300° = 60°
  3. cos 300° = 1/2cosine is positive in quadrant IV

Both reduce to an angle between 0° and 360° first. A whole turn never changes a trig value, so adding or subtracting 360° is always safe.

An angle in radians

8. sin 5π/6worked example
  1. 5π/6 = 150°multiply by 180/π to compare with the degree table
  2. 150°quadrant II, reference angle 30°
  3. sin 150° = 1/2sine is positive in quadrant II

In radians the same reduction works with π in place of 180°: the reference angle of 5π/6 is π − 5π/6 = π/6. Exact values exist here because 5π/6 is a standard angle.

Angles on an axis

At 180° the point on the unit circle is (−1, 0), so sin 180° = 0 and cos 180° = −1. Then tan 180° = 0 ÷ (−1) = 0. At 270° the point is (0, −1), so cos 270° = 0 and tan 270° is undefined. On an axis there is no quadrant, so the values come straight from the coordinates.

Checking every result the same way

Two checks catch nearly every mistake: the size must match the reference angle, and the sign must match the quadrant.

ExampleReference angleQuadrantSign check
1 sin 120°60°IIsin positive ✓
3 cos 225°45°IIIcos negative ✓
5 tan 300°60°IVtan negative ✓

This is how the Trig Calculator on this site is checked too: every value is compared with an independent computation, and the sign has to follow the quadrant.

What all eight worked examples have in common

Every one of these worked examples is the same step by step routine: reduce the angle, name the quadrant, find the reference angle, read its value from the standard angles, and attach the sign. The eight examples cover degrees and radians, positive and negative angles, and all four quadrants, but never a new technique.

Whenever you finish one, check your answer with the two questions above. If the sign agrees with the quadrant and the size agrees with the reference angle, the value is almost certainly right.

Questions about trig examples

What is sin 120°?

√3/2. The reference angle is 60°, and sine is positive in the second quadrant.

What is tan 300°?

−√3. The reference angle is 60°, so the size is √3, and tangent is negative in the fourth quadrant.

How do I find the trig value of a negative angle like −60°?

Add 360° to get 300°, then use its quadrant and reference angle: cos(−60°) = cos 300° = 1/2.

What is sin 5π/6?

1/2. 5π/6 is 150°, whose reference angle is 30°, and sine is positive in the second quadrant.

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