Math · Precalculus · Worked example

Evaluate trig at negative and large angles

Find the exact values of sin(−π/3), cos(11π/4) and tan(7π/6).

A negative angle

−π/3 turns clockwise into quadrant IV. Sine is odd, so sin(−π/3) = −sin(π/3).

sin (- \frac{π}{3}) = - \frac{\sqrt{3}}{2}

Remove whole turns

11π/4 is more than 2π. Subtracting one turn leaves the coterminal angle 3π/4, in quadrant II with reference angle π/4, where cosine is negative.

\begin{array}{l} \frac{11 π}{4} - 2 π = \frac{3 π}{4} \\ cos \frac{11 π}{4} = - \frac{\sqrt{2}}{2} \end{array}

Quadrant III

7π/6 has reference angle π/6, and both coordinates are negative in quadrant III: the point is (−√3/2, −1/2). The two signs cancel in the ratio, so the tangent is positive.

tan \frac{7 π}{6} = \frac{\sqrt{3}}{3}

Result

sin(−π/3) = −√3/2, cos(11π/4) = −√2/2 and tan(7π/6) = √3/3.

Your turn

Find the exact values of cos(−5π/6) and sin(13π/6).

Show the answer and explanation

cos(−5π/6) = −√3/2 and sin(13π/6) = 1/2.

Cosine is even, so cos(−5π/6) = cos(5π/6) = −√3/2 (quadrant II, reference angle π/6). And 13π/6 − 2π = π/6, so sin(13π/6) = sin(π/6) = 1/2.

\begin{array}{l} cos (- \frac{5 π}{6}) = - \frac{\sqrt{3}}{2} \\ sin \frac{13 π}{6} = \frac{1}{2} \end{array}

Keep exploring

In Graph, follow y = sin x to x = −π/3 ≈ −1.047: the height is about −0.866, which is −√3/2.

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