A reference angle is the acute angle between the terminal side and the x-axis. It is always between 0° and 90°.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding it
In quadrant II subtract from 180°. In quadrant III subtract 180°. In quadrant IV subtract from 360°.
The size is the same
The reference angle gives the size of the ratio. Only the sign is left to decide, and the quadrant decides it.
Which are positive
Sine is positive where y is positive, quadrants I and II. Cosine is positive where x is positive, quadrants I and IV. Tangent is y ÷ x, positive where the signs agree, quadrants I and III.
An example
For 150°, the reference angle is 30°, and quadrant II makes sine positive. So sin 150° = sin 30° = 0.5.
The reference angle
The reference angle is the acute angle between the terminal side and the x-axis. Every trigonometric value for any angle equals the value for its reference angle, up to a sign determined by the quadrant.
Finding it in each quadrant
In the second quadrant it is 180° − θ; in the third, θ − 180°; in the fourth, 360° − θ. Always measure to the x-axis, never the y-axis — measuring to the wrong axis is the standard error.
The two-step procedure
Find the reference angle to get the size, then use the quadrant to get the sign. Separating magnitude from sign makes every angle computable from the handful of acute values you know.
Coterminal angles
Angles differing by full turns land on the same point: 30°, 390° and −330° all have identical trigonometric values. Reducing an angle to the range 0° to 360° before anything else keeps the work manageable.
Step 2: Try It Yourself
Tap and try it out.
- Angle150° = 5π/6 rad
- x-coordinate-0.866
- y-coordinate0.500
- cos 150°-0.866
- sin 150°0.500
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Marcus Evaluate cos 210°
Marcus needs cos 210° without a calculator.
- Step 1
He locates 210°: past 180°, so it lands in quadrant III.
Step 4: Your Turn
Practice makes it stick.
The Reference
Problem 1 of 2
What is the reference angle for 240°, in degrees?
degrees
The Sign
Problem 2 of 2
In which quadrant are sine and cosine both negative?
Any Angle at All
1 of 8
Reference angle for 150°, in degrees?
2 of 8
Reference angle for 300°, in degrees?
3 of 8
Reference angle for 225°, in degrees?
4 of 8
sin 150° as a decimal?
5 of 8
sin 210° as a decimal?
6 of 8
In which quadrants is tangent positive?
7 of 8
Put these angles in order by size of reference angle, smallest first.
- 1200°
- 2310°
- 3225°
- 4170°
8 of 8
cos 300° as a decimal?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is the reference angle for 200°, in degrees?
Question 2 of 2
What is sin 330° as a decimal?
What You Learned
- The reference angle is the acute angle to the x-axis.
- It sets the size of the ratio; the quadrant sets the sign.
- Sine follows y, cosine follows x, tangent is positive when they agree.