Sine and cosine on the unit circle: special angles

About this lesson

Integrated Math III › Trigonometric functions

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Sine and cosine on the unit circle: special angles

Warm-up

Three quick questions from earlier lessons.

Discover

On the circle of radius 1 centered at the origin, start at (1,0)(1, 0) and turn by an angle θ\theta. You reach a point. At θ=π2\theta = \dfrac{\pi}{2}, which point is it?

  1. (0,1)(0, 1)
  2. (1,0)(1, 0)

Key idea

The unit circle has radius 1. Turning by θ\theta from (1,0)(1, 0) lands on a point we name with sine and cosine.

The point at angle θ\theta is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta): cosine is the xx-coordinate, sine is the yy-coordinate. Special angles: cos⁡π6=32\cos\dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}, sin⁡π6=12\sin\dfrac{\pi}{6} = \dfrac{1}{2}; cos⁡π4=sin⁡π4=22\cos\dfrac{\pi}{4} = \sin\dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}; cos⁡π3=12\cos\dfrac{\pi}{3} = \dfrac{1}{2}, sin⁡π3=32\sin\dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}.

Worked example

Find sin⁡π3\sin\dfrac{\pi}{3} and cos⁡π3\cos\dfrac{\pi}{3}.

The point is (12,32)\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)
π/3 = 60°: the point is high and close to the y-axis.
cos⁡π3=12\cos\dfrac{\pi}{3} = \dfrac{1}{2}
Cosine is the x-coordinate.
sin⁡π3=32\sin\dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}
Sine is the y-coordinate.

Your turn, guided

Your turn: θ=π6\theta = \dfrac{\pi}{6}.

The point is (32,12)\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)
π/6 = 30°: the point is low and far to the right.
sin⁡π6=12\sin\dfrac{\pi}{6} = \dfrac{1}{2}
The y-coordinate.

Practice ★

  1. Find the exact value of cos⁡(π6)\cos\left(\frac{\pi}{6}\right).
  2. Find the exact value of cos⁡(π3)\cos\left(\frac{\pi}{3}\right).
  3. Find the exact value of cos⁡(π2)\cos\left(\frac{\pi}{2}\right).
  4. Find the exact value of cos⁡(0)\cos\left(0\right).

Watch out!

Practice ★★

  1. Find the exact value of sin⁡(π2)\sin\left(\frac{\pi}{2}\right).
  2. Find the exact value of sin⁡(π6)\sin\left(\frac{\pi}{6}\right).
  3. Find the exact value of sin⁡(π3)\sin\left(\frac{\pi}{3}\right).
  4. Find the exact value of sin⁡(0)\sin\left(0\right).
  1. Match each angle with its point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) on the unit circle.
    1. π6\dfrac{\pi}{6}
    2. π4\dfrac{\pi}{4}
    3. π3\dfrac{\pi}{3}
    4. π2\dfrac{\pi}{2}
    1. (32,12)\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)
    2. (22,22)\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)
    3. (12,32)\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)
    4. (0,1)(0, 1)

Exit ticket

Three questions to check you have got it.

  1. Find the exact value of sin⁡(π2)\sin\left(\frac{\pi}{2}\right).
  2. Find the exact value of sin⁡(π6)\sin\left(\frac{\pi}{6}\right).
  3. Find the exact value of sin⁡(π3)\sin\left(\frac{\pi}{3}\right).

Summary card

The unit circle

  • The point at angle θ\theta is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).
  • π6\dfrac{\pi}{6}: (32,12)\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right); π4\dfrac{\pi}{4}: (22,22)\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right); π3\dfrac{\pi}{3}: (12,32)\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right).

Watch out: cosine is xx, sine is yy.

End of the lesson

Next lesson: Reference angles and signs in each quadrant Back to the chapter

Answers: Sine and cosine on the unit circle: special angles

A

  1. 32\frac{\sqrt{3}}{2}
  2. 12\frac{1}{2}
  3. 00
  4. 11

B

  1. 11
  2. 12\frac{1}{2}
  3. 32\frac{\sqrt{3}}{2}
  4. 00

C

  1. π6\dfrac{\pi}{6} → (32,12)\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) · π4\dfrac{\pi}{4} → (22,22)\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right) · π3\dfrac{\pi}{3} → (12,32)\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) · π2\dfrac{\pi}{2} → (0,1)(0, 1)

D

  1. 11
  2. 12\frac{1}{2}
  3. 32\frac{\sqrt{3}}{2}

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