Grade by Grade
D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Related Acute Angles and Exact Values from 0° to 360°

D1.2 D1.1About 70 minutes
Learning goals
  • I can find the related acute angle (reference angle) for any angle from 0° to 360°.
  • I can use the related acute angle and the CAST rule to find exact values of sin, cos and tan for angles such as 150°, 225° and 300°.
  • I can use the unit circle to explain why angles in different quadrants share the same ratios up to sign.
  • I can check exact values with a calculator in degree mode.

Words to know

related acute angle
The acute angle β between the terminal arm of θ and the x-axis. Also called the reference angle. It is always between 0° and 90°.
unit circle
The circle with centre (0, 0) and radius 1. The point where the terminal arm of θ meets it is (cos θ, sin θ).
principal angle
The angle θ between 0° and 360° measured counterclockwise from the positive x-axis.

Four angles, one triangle

What do 30°, 150°, 210° and 330° have in common? Each terminal arm makes a 30° angle with the x-axis. If you drop a perpendicular to the x-axis from each one, you get the same 30-60-90 triangle, just flipped into a different quadrant. So their sines and cosines have the same size; only the signs change. This lesson turns that idea into a quick method for exact values.

The related acute angle

The related acute angle β (beta) is the acute angle between the terminal arm and the x-axis (never the y-axis).

  • Quadrant I: β = θ
  • Quadrant II: β = 180° − θ
  • Quadrant III: β = θ − 180°
  • Quadrant IV: β = 360° − θ
βQuadrant Iβ = θe.g. θ = 40°, β = 40°βQuadrant IIβ = 180° − θe.g. θ = 150°, β = 30°βQuadrant IIIβ = θ − 180°e.g. θ = 220°, β = 40°βQuadrant IVβ = 360° − θe.g. θ = 320°, β = 40°
The shaded wedge in each quadrant is the related acute angle β, measured to the nearest part of the x-axis. The small arrow shows θ, measured from the positive x-axis.

The method

To find a ratio of any angle θ:

  1. Find the quadrant of θ.
  2. Find the related acute angle β.
  3. Find the ratio of β (exactly, from the special triangles, or with a calculator).
  4. Attach the sign from CAST.

In short: trig ratio of θ = ± (same ratio of β).

Worked example

Example 1: sin 150°

Find the exact value of sin 150°.

  1. 150° is in quadrant II.
  2. β = 180° − 150° = 30°.
  3. sin 30° = 1/2.
  4. In quadrant II sine is positive (S in CAST).

sin 150° = 1/2

Worked example

Example 2: cos 225°

Find the exact value of cos 225°.

  1. 225° is in quadrant III.
  2. β = 225° − 180° = 45°.
  3. cos 45° = √2/2.
  4. In quadrant III only tangent is positive, so cosine is negative.

cos 225° = −√2/2

Worked example

Example 3: tan 300°

Find the exact value of tan 300°.

  1. 300° is in quadrant IV, and β = 360° − 300° = 60°.
  2. tan 60° = √3.
  3. In quadrant IV only cosine is positive, so tangent is negative.

tan 300° = −√3

Why it works: the unit circle

On the unit circle (radius 1), r = 1, so sin θ = y/1 = y and cos θ = x/1 = x. The point on the terminal arm is (cos θ, sin θ).

The point for 30° is (√3/2, 1/2). Reflect it in the y-axis and you get the point for 150°: (−√3/2, 1/2). Reflect that in the x-axis for 210°: (−√3/2, −1/2). Reflect back across the y-axis for 330°: (√3/2, −1/2). The coordinates keep the same size, and the signs change exactly as CAST predicts.

xy30°: (√3/2, 1/2)150°: (−√3/2, 1/2)210°: (−√3/2, −1/2)330°: (√3/2, −1/2)All four points are reflectionsof the 30° point in the axes.Same size coordinates,different signs:the signs follow CAST.
On the unit circle, each point is (cos θ, sin θ). The points for 30°, 150°, 210° and 330° are mirror images of each other.
Worked example

Example 4: Checking with technology

Use a calculator to compare sin 140° with sin 40°, and cos 140° with cos 40°. Explain the results.

  1. sin 140° ≈ 0.6428 and sin 40° ≈ 0.6428: equal.
  2. cos 140° ≈ −0.7660 and cos 40° ≈ 0.7660: same size, opposite signs.
  3. 140° is in quadrant II with β = 40°. Sine is positive there and cosine is negative.

sin 140° = sin 40° and cos 140° = −cos 40°, exactly as the method predicts. Dynamic geometry software (such as GeoGebra or Desmos) lets you drag a point around the unit circle and watch this happen.

Worked example

Example 5: A longer expression

Find the exact value of sin 120° cos 330° − tan 225°.

  1. sin 120°: quadrant II, β = 60°, positive. sin 120° = √3/2.
  2. cos 330°: quadrant IV, β = 30°, positive. cos 330° = √3/2.
  3. tan 225°: quadrant III, β = 45°, positive. tan 225° = 1.
  4. (√3/2)(√3/2) − 1 = 3/4 − 1 = −1/4.

−1/4

θ120°135°150°210°225°240°300°315°330°
β60°45°30°30°45°60°60°45°30°
sin θ√3/2√2/21/2−1/2−√2/2−√3/2−√3/2−√2/2−1/2
cos θ−1/2−√2/2−√3/2−√3/2−√2/2−1/21/2√2/2√3/2
tan θ−√3−1−√3/3√3/31√3−√3−1−√3/3
Exact values for the angles related to 30°, 45° and 60°. Don't memorize this table: rebuild any entry with the four-step method.
Just for fun

Riddle: Which three angles between 90° and 360° are always on good terms with 30°?

150°, 210° and 330°. They're related.

Hands-on activity

Build a unit circle you can hold

You need: a paper plate or a sheet of paper, a ruler, a protractor, a pencil and a calculator. (A paper plate is easy to turn.)

  1. Draw axes through the centre. Draw (or trace) a circle with radius 10 cm. Let 10 cm stand for 1 unit.
  2. Use the protractor to mark 30°, 150°, 210° and 330° on the circle.
  3. Measure the x- and y-coordinates of each point in centimetres and divide by 10.
  4. Compare with the exact values (√3/2 ≈ 0.87 and 1/2 = 0.5) and with the signs from CAST.
  5. Repeat for 45°, 135°, 225° and 315°.

What to look for: every point in the first set should be about (±0.87, ±0.5) and every point in the second set about (±0.71, ±0.71), with signs that match the quadrant.

Your turn

1

Find the exact value of sin 135°.

2

Find the exact value of cos 120°.

3

Find the exact value of tan 210°.

4

Find the exact value of sin 300°.

5

Find the exact value of cos 315°.

6

Find the exact value of tan 150°.

7

What is the related acute angle for 255°?

More practice: warm-up

1

Related acute angle for 160°?

2

Related acute angle for 200°?

3

Related acute angle for 340°?

4

Related acute angle for 95°?

5

Is sin 250° positive or negative?

More practice: on your own

1

Find the exact value of sin 240° + cos 150°.

2

Find the exact value of cos²135° + sin 330°.

3

Find the exact value of tan 120° × tan 300°.

4

Use a calculator to show that sin 200° = −sin 20°. Explain why using the related acute angle.

5

Find the exact coordinates of the point where the terminal arm of 225° meets the unit circle.

Challenge

1

For an acute angle θ, explain using the unit circle why sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ.

2

Find the exact value of (2 sin 210° + cos 180°) ÷ tan 135°.

3

A point P is on the terminal arm of 300°, 8 units from the origin. Find the exact coordinates of P.

Check your understanding

Quiz

1.

What is the related acute angle of 210°?

2.

What is the exact value of sin 150°?

3.

What is the exact value of cos 225°?

4.

What is the exact value of tan 300°?

5.

On the unit circle, the point on the terminal arm of θ is:

6.

Which of these equals −cos 40°?

For parents and teachers

Big idea (D1.2, D1.1): any angle from 0° to 360° has a related acute angle β between its terminal arm and the x-axis. Its ratios are the ratios of β with a sign from CAST. Combined with the special triangles from lesson 1, this gives exact values for angles like 135°, 240° and 330°.

How to help: Name an angle such as 240° and ask four questions: Which quadrant? What is β? What is the ratio of β? What sign? Then check the decimal on a calculator (degree mode).

Common difficulties:

  • Measuring β to the y-axis instead of the x-axis (for 120°, β is 60°, not 30°).
  • Getting the size right but the sign wrong. A quick sketch prevents this.
  • Forgetting that (√2/2)² = 1/2 when an expression has a square.