Aim for FivePrecalculus

Unit 3 · Topic 3.3

3.3 Sine and Cosine Function Values

For angles that are multiples of π/6 and π/4, you can find exact sine and cosine values with two special right triangles and the signs of each quadrant. These exact values come up constantly on no-calculator questions.

Key terms

  • special right triangles
  • reference angle
  • quadrant
  • exact value
  • coordinates on the unit circle

Points on any circle

If the terminal ray of θ meets a circle of radius r centered at the origin at P, then P = (r cos θ, r sin θ). This follows from cos θ = x/r and sin θ = y/r.

So a point 6 units from the origin at angle π/3 is (6 cos(π/3), 6 sin(π/3)).

The two special triangles

Isosceles right triangle (45°-45°-90°): the legs are equal. With hypotenuse 1, each leg is 1/√2 = √2/2. So for θ = π/4, the point on the unit circle is (√2/2, √2/2).

Half of an equilateral triangle (30°-60°-90°): with hypotenuse 1, the short leg is 1/2 (opposite the 30° angle) and the long leg is √3/2 (opposite the 60° angle). So the unit-circle point for π/6 is (√3/2, 1/2), and for π/3 it's (1/2, √3/2).

θ0π/6π/4π/3π/2
cos θ1√3/2√2/21/20
sin θ01/2√2/2√3/21
tan θ0√3/31√3undefined

Reference angles and signs

For an angle outside the first quadrant, find its reference angle: the acute angle between the terminal ray and the x-axis. The sine and cosine have the same size as for the reference angle; only the signs change, depending on the quadrant.

  • Quadrant II: reference angle π − θ. Example: 5π/6 has reference angle π/6.
  • Quadrant III: reference angle θ − π. Example: 4π/3 has reference angle π/3.
  • Quadrant IV: reference angle 2π − θ. Example: 7π/4 has reference angle π/4.

Angles on the axes

When the terminal ray lies on an axis, read the point directly: θ = 0 gives (1, 0), θ = π/2 gives (0, 1), θ = π gives (−1, 0), and θ = 3π/2 gives (0, −1). So cos π = −1 and sin(3π/2) = −1.

A quick memory check: as θ goes from 0 to π/2, sine increases from 0 to 1 (0, 1/2, √2/2, √3/2, 1) while cosine runs through the same values in reverse.

For an angle bigger than 2π or less than 0, add or subtract 2π until it lands in [0, 2π). For example, 19π/6 − 2π = 7π/6, which is in quadrant III with reference angle π/6, so sin(19π/6) = −1/2.

Exact tangent values come from dividing: tan(π/6) = (1/2)/(√3/2) = 1/√3, which is usually written √3/3. Then attach the quadrant's sign, as with sine and cosine.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1

    A quadrant II angle

    Find the exact values of sin(5π/6), cos(5π/6) and tan(5π/6).

    Show the solution
    1. Step 1: 5π/6 is in quadrant II, and its reference angle is π − 5π/6 = π/6.
    2. Step 2: For π/6: sin = 1/2 and cos = √3/2.
    3. Step 3: In quadrant II, sine is positive and cosine is negative: sin(5π/6) = 1/2, cos(5π/6) = −√3/2.
    4. Step 4: tan(5π/6) = (1/2)/(−√3/2) = −1/√3 = −√3/3.

    Answer: sin(5π/6) = 1/2, cos(5π/6) = −√3/2, tan(5π/6) = −√3/3.

  2. Example 2

    A point on a larger circle

    The terminal ray of θ = 2π/3 meets a circle of radius 6 centered at the origin at P. Find the exact coordinates of P.

    Show the solution
    1. Step 1: P = (6 cos(2π/3), 6 sin(2π/3)).
    2. Step 2: 2π/3 is in quadrant II with reference angle π/3, so cos(2π/3) = −1/2 and sin(2π/3) = √3/2.
    3. Step 3: P = (6 · (−1/2), 6 · √3/2) = (−3, 3√3).

    Answer: P = (−3, 3√3).

  3. Example 3

    Trap: signs in quadrants III and IV

    Find cos(−3π/4) and sin(4π/3).

    Show the solution
    1. Step 1: −3π/4 means 3π/4 clockwise, which lands in quadrant III. Its reference angle is π/4. In quadrant III, cosine is negative: cos(−3π/4) = −√2/2.
    2. Step 2: 4π/3 is in quadrant III with reference angle π/3. Sine is negative there: sin(4π/3) = −√3/2.
    3. Step 3: A common slip is to give the reference-angle value with a positive sign. Always place the angle first, then attach the sign.

    Answer: cos(−3π/4) = −√2/2 and sin(4π/3) = −√3/2.

Common mistakes

  • Swapping the values for π/6 and π/3. Remember sin(π/6) = 1/2: a small angle has a small height.
  • Forgetting the sign after finding the reference-angle value.
  • Placing negative angles counterclockwise. A negative angle rotates clockwise from the positive x-axis.

On the exam

  • Part A of multiple choice is no-calculator, so expect exact values like cos(7π/6) or sin(−π/4), sometimes inside a larger expression.
  • The periodic-modeling free-response question is no-calculator, and often needs exact values like sin(π/6) to evaluate a model.

Connected topics

Videos

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  • Exact Values of Trigonometric Functions EXPLAINED [AP Precalculus Topic 3.3] #apprecalculus

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Check yourself

4 questions on 3.3 Sine and Cosine Function Values. Pick an answer to see if you got it, and why.

Question 1 of 4

A circle of radius 10 is centered at the origin. An angle of 2π/3 radians in standard position has a terminal ray that intersects the circle at point P. What are the coordinates of P?

Question 2 of 4

Which of the following is equal to sin(π/5)?

Question 3 of 4

What is the exact value of cos(5π/6) + sin(4π/3)?

Question 4 of 4

For 0 ≤ θ < 2π, the terminal ray of θ meets the unit circle at (−1/2, −√3/2). What is θ?

0 of 4 answered