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Learn · Mathematics · Trigonometry

Angles beyond 90°: reference angles and signs

How 150° borrows its value from 30°, which functions are positive in each quadrant, and the identities for 180° − a, 180° + a and 360° − a.

High school Lesson 3 of 9 Interactive figure Practice

First, the rules

Borrow the value, fix the sign

Every angle on the circle has a reference angle between 0° and 90°. The function has the same size at both; only the sign depends on the quadrant.

★ Second quadrant (90° to 180°)
Examples 1, 2
sin(180° − a) = sin a, cos(180° − a) = −cos a, tan(180° − a) = −tan a
Third quadrant (180° to 270°)
Example 2
sin(180° + a) = −sin a, cos(180° + a) = −cos a, tan(180° + a) = tan a
Fourth quadrant (270° to 360°)
Example 2
sin(360° − a) = −sin a, cos(360° − a) = cos a, tan(360° − a) = −tan a
Negative angles
sin(−a) = −sin a, cos(−a) = cos a, tan(−a) = −tan a
Clockwise by a is the same point as 360° − a.

Which quadrant makes what positive

I · all +II · sin +III · tan, cot +IV · cos +

Counter-clockwise from the first quadrant: All, Sin, Tan, Cos — "All Students Take Calculus". Cot follows tan.

★ Reference angle in each quadrant
II: 180° − θ · III: θ − 180° · IV: 360° − θ

See it

150° is 30° in disguise

Press 150° and watch: the arm is the mirror image of 30°, the height (sine) is identical, the horizontal distance (cosine) flips to the left — negative. The thin grey arm is the reference angle.

Step by step

Worked examples

1

A quotient of four values

reference angles

Evaluate sin 150° · cos 135°cos 120° · sin 135°.

Ask yourself: Which quadrant is 150° in? What are the signs of sine and cosine there?

Show the solution
  1. sin 150° = sin 30° = 12 (second quadrant, sine positive).
  2. cos 135° = −cos 45° = −√22 (second quadrant, cosine negative).
  3. cos 120° = −cos 60° = −12.
  4. sin 135° = sin 45° = √22.
  5. Numerator: 12 · (−√22) = −√24. Denominator: (−12) · √22 = −√24. They are equal.

Answer 1

2

Values in every quadrant

signs

Find sin 210°, cos 300°, tan 225° and cot 120°.

Ask yourself: Write each angle as 180° ± a or 360° − a first, then decide the sign from the quadrant.

Show the solution
  1. 210° = 180° + 30°, third quadrant, sine negative: sin 210° = −12.
  2. 300° = 360° − 60°, fourth quadrant, cosine positive: cos 300° = cos 60° = 12.
  3. 225° = 180° + 45°, third quadrant, tangent positive: tan 225° = tan 45° = 1.
  4. 120° = 180° − 60°, second quadrant, cotangent negative: cot 120° = −cot 60° = −√33.

Answer −12, 12, 1, −√33

Practice: angles beyond 90°

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Only angles from 90° to 360° here — the reference angle and the sign are the whole game. Your best score is kept in this browser only.

About this lesson

Evaluate trigonometric functions of obtuse and reflex angles with reference angles: sin(180° − a) = sin a, cos(180° − a) = −cos a and the rest of the family. Learn the quadrant sign rule with a mnemonic, see it on the unit circle, and practise exact values such as sin 150°, cos 135° and tan 240°.

Part of Learn mathematics · unit Trigonometry. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.