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B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Exact Values for Special Angles in Radians: The Unit Circle

B1.4About 70 minutes
Learning goals
  • I can rebuild the special triangles with angles π/6, π/4 and π/3 and read their exact ratios.
  • I can use the related acute angle and the CAST rule to find exact values for multiples of π/6 and π/4 between 0 and 2π.
  • I can give the exact coordinates of the special points on the unit circle.
  • I can evaluate expressions with special-angle ratios exactly.

Words to know

special angles (in radians)
0, π/6, π/4, π/3, π/2 and their multiples up to 2π. Their ratios have exact values with simple radicals.
related acute angle (β)
The acute angle between the terminal arm and the x-axis. It is always between 0 and π/2.
exact value
A value written with integers, fractions and radicals, such as −√3/2, not a rounded decimal.
quadrantal angle
An angle whose terminal arm lies on an axis: 0, π/2, π, 3π/2, 2π, and so on.

Exact values, now in radians

In Grade 11 you used the two special triangles to find exact values for 30°, 45° and 60°, and then for all their relatives up to 360°. Today you will do the same thing with radian labels, so that you can write answers like cos(5π/6) = −√3/2 without a calculator. These exact values appear everywhere in the rest of this unit, in the unit Trigonometric Identities and Equations, and in calculus.

The special triangles in radians:

  • half a square: angles π/4, π/4, π/2, sides 1, 1, √2
  • half an equilateral triangle: angles π/6, π/3, π/2, sides 1, √3, 2 (the side 1 is opposite π/6)
θ0π/6π/4π/3π/2
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ0√3/31√3undefined
Exact values in the first quadrant. The sines run √0/2, √1/2, √2/2, √3/2, √4/2; the cosines are the same list backwards.

The unit circle

Scale each special triangle so that its hypotenuse is 1 and put it in standard position. The vertex on the circle is the point (cos θ, sin θ). For example, the 30-60-90 triangle with hypotenuse 1 has legs √3/2 and 1/2, so the point for θ = π/6 is (√3/2, 1/2).

Now reflect these points in the axes. Each reflection keeps the same numbers but can change the signs. That gives exact coordinates for every multiple of π/6 and π/4 around the circle.

0(1, 0)π/6(√3/2, 1/2)π/4(√2/2, √2/2)π/3(1/2, √3/2)π/2(0, 1)2π/3(−1/2, √3/2)3π/4(−√2/2, √2/2)5π/6(−√3/2, 1/2)π(−1, 0)7π/6(−√3/2, −1/2)5π/4(−√2/2, −√2/2)4π/3(−1/2, −√3/2)3π/2(0, −1)5π/3(1/2, −√3/2)7π/4(√2/2, −√2/2)11π/6(√3/2, −1/2)
The unit circle. Each point is (cos θ, sin θ). Read cosine from the first coordinate and sine from the second; tan θ = sin θ ÷ cos θ.

The related acute angle in radians

To find an exact value without the whole picture:

  1. Find the quadrant of θ.
  2. Find the related acute angle β (the angle to the x-axis, never the y-axis):
  • quadrant II: β = π − θ
  • quadrant III: β = θ − π
  • quadrant IV: β = 2π − θ
  1. Find the ratio of β from the special triangles.
  2. Attach the sign from CAST.

Shortcut: look at the denominator. Angles like 5π/6, 7π/6, 11π/6 have β = π/6. Angles like 3π/4, 5π/4, 7π/4 have β = π/4. Angles like 2π/3, 4π/3, 5π/3 have β = π/3.

π/6(√3/2, 1/2)π − π/6 = 5π/6(−√3/2, 1/2)π + π/6 = 7π/6(−√3/2, −1/2)2π − π/6 = 11π/6(√3/2, −1/2)ββββ
The four angles with related acute angle β = π/6 make a rectangle on the unit circle. The coordinates have the same sizes; only the signs change from quadrant to quadrant.
Worked example

Example 1: A quadrant II angle

Find the exact value of sin(5π/6).

  1. 5π/6 is between π/2 and π: quadrant II.
  2. β = π − 5π/6 = π/6.
  3. sin(π/6) = 1/2, and sine is positive in quadrant II.

sin(5π/6) = 1/2

Worked example

Example 2: A quadrant III angle

Find the exact value of cos(4π/3).

  1. 4π/3 is between π and 3π/2: quadrant III.
  2. β = 4π/3 − π = π/3.
  3. cos(π/3) = 1/2, but cosine is negative in quadrant III.

cos(4π/3) = −1/2

Worked example

Example 3: A quadrant IV tangent

Find the exact value of tan(7π/4).

  1. 7π/4 is between 3π/2 and 2π: quadrant IV.
  2. β = 2π − 7π/4 = π/4, and tan(π/4) = 1.
  3. Tangent is negative in quadrant IV.

tan(7π/4) = −1

Worked example

Example 4: An expression

Find the exact value of sin(2π/3) cos(11π/6) − cos(3π/2).

  1. sin(2π/3): quadrant II, β = π/3, sine positive: √3/2.
  2. cos(11π/6): quadrant IV, β = π/6, cosine positive: √3/2.
  3. cos(3π/2): the point is (0, −1), so cosine is 0.
  4. Substitute: (√3/2)(√3/2) − 0 = 3/4.

3/4

Just for fun

The unit circle is the calmest circle in mathematics. Whatever happens, it always keeps its radius at 1.

Hands-on activity

Paper-triangle unit circle

You need: your radian clock (paper plate) from the lesson Converting Between Radians and Degrees, or a new circle of radius 10 cm drawn with a compass or traced from a bowl; a ruler, a protractor, a pencil, a sheet of stiff paper and scissors (be careful with them).

  1. On the stiff paper, draw a right triangle with hypotenuse 10 cm and angles of 30° (π/6) and 60° (π/3). Its legs should measure about 5.0 cm and 8.7 cm (10 × 1/2 and 10 × √3/2). Cut it out.
  2. Draw and cut out a right isosceles triangle with hypotenuse 10 cm. Each leg should be about 7.1 cm (10 × √2/2).
  3. Place the 30-60-90 triangle in quadrant I with its π/6 corner at the centre and its long leg on the x-axis. Its top corner should land on the π/6 mark on your circle. Mark the point.
  4. Flip the triangle into quadrants II, III and IV so its corner lands on 5π/6, 7π/6 and 11π/6. Write the coordinates at each point, with signs.
  5. Repeat with the triangle stood on its short leg (for π/3, 2π/3, 4π/3, 5π/3) and with the isosceles triangle (for the multiples of π/4).

What to look for: with a radius of 10 cm, every coordinate is 10 times the unit-circle value, for example (−8.7, 5.0) at 5π/6. You should end up with all 16 points from the unit circle diagram, and you will see that only the signs change from quadrant to quadrant.

Your turn

1

Find the exact value of sin(2π/3).

2

Find the exact value of cos(3π/4).

3

Find the exact value of tan(5π/6).

4

Find the exact value of sin(7π/6).

5

Find the exact value of cos(5π/3).

6

Find the exact value of tan(5π/4).

7

Give the exact coordinates of the point on the unit circle for θ = 11π/6.

More practice: warm-up

1

sin(π/6) = ?

2

cos(π/4) = ?

3

tan(π/3) = ?

4

cos π = ?

5

sin(π/2) = ?

More practice: on your own

1

Find the exact value of cos(7π/6) · sin(4π/3).

2

Find the exact value of sin²(3π/4) + cos²(3π/4).

3

Find the exact value of tan(2π/3) + tan(π/3).

4

Which angles between 0 and 2π have related acute angle π/4 and a negative cosine?

5

Find the exact value of 2 sin(π/6) cos(π/6). Which special-angle ratio is it equal to?

Challenge

1

Find the exact value of sin(5π/4) cos(π/3) − cos(5π/4) sin(π/3).

2

A point on the unit circle has y-coordinate −1/2 and a positive x-coordinate. Find the angle θ (0 ≤ θ < 2π) and the exact coordinates.

3

Show that cos(π/3 + π/6) is not equal to cos(π/3) + cos(π/6).

Check your understanding

Quiz

1.

What is the exact value of cos(5π/6)?

2.

What is the exact value of sin(7π/4)?

3.

What is the exact value of tan(4π/3)?

4.

What is the related acute angle of 5π/3?

5.

Which point is on the unit circle at θ = 3π/4?

6.

What is sin(3π/2)?

For parents and teachers

Big idea (B1.4): every exact value comes from two triangles (1, 1, √2 and 1, √3, 2), the related acute angle and the CAST signs. The unit circle collects them all in one picture.

How to help: Ask your teen to draw the two special triangles from memory, then name an angle such as 4π/3 and ask: "Which quadrant? What is β? Which sign?" Let them check with a calculator in radian mode (for example, cos(4π/3) should give −0.5).

Common difficulties:

  • Measuring β from the y-axis instead of the x-axis.
  • Getting the size right but the sign wrong (CAST).
  • Mixing up which of π/6 and π/3 has sine 1/2 (sine of the smaller angle is the smaller value).
  • Trying to memorize all 16 points instead of using the pattern.