Math · Precalculus · Worked example

Find exact trig values with a reference angle

Find the exact values of sin θ, cos θ and tan θ for θ = 2π/3.

θ = \frac{2 π}{3}

Locate the angle

2π/3 is two thirds of a half-turn: more than π/2 and less than π, so its terminal side is in quadrant II. In degrees it is 120°.

\frac{2 π}{3} \cdot \frac{180^{\circ}}{π} = 120^{\circ}

Find the reference angle

In quadrant II the reference angle is π − θ.

π - \frac{2 π}{3} = \frac{π}{3}

Use the values at π/3 with quadrant II signs

At π/3 the point is (1/2, √3/2). In quadrant II x is negative and y is positive, so the point at 2π/3 is (−1/2, √3/2).

\begin{array}{l} cos \frac{2 π}{3} = - \frac{1}{2} \\ sin \frac{2 π}{3} = \frac{\sqrt{3}}{2} \end{array}

Divide for the tangent

Tangent is sine divided by cosine: (√3/2) ÷ (−1/2) = −√3.

tan \frac{2 π}{3} = - \sqrt{3}

Result

cos(2π/3) = −1/2, sin(2π/3) = √3/2 and tan(2π/3) = −√3.

Your turn

Find the exact values of sin θ, cos θ and tan θ for θ = 5π/4.

Show the answer and explanation

sin(5π/4) = −√2/2, cos(5π/4) = −√2/2 and tan(5π/4) = 1.

5π/4 is in quadrant III with reference angle 5π/4 − π = π/4. Both coordinates are negative there, so the point is (−√2/2, −√2/2), and their ratio is 1.

\begin{array}{l} sin \frac{5 π}{4} = - \frac{\sqrt{2}}{2} \\ cos \frac{5 π}{4} = - \frac{\sqrt{2}}{2} \\ tan \frac{5 π}{4} = 1 \end{array}

Keep exploring

Open the Unit Circle and choose the angle 2π/3: it shows sin θ = √3/2, cos θ = −1/2 and tan θ = −√3. Choose 4π/3 and the sine and cosine are both negative.

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