SL 3.5—Unit circle and exact values

Use the unit circle for exact trigonometric values.

On the unit circle, cosθ is x and sinθ is y; quadrant signs determine exact values and tanθ=sinθ/cosθ.

Worked example

At 5π/6, sinθ=1/2 and cosθ=−√3/2, so tanθ=−1/√3.

Reference angles provide magnitude; the quadrant supplies the sign.

Do not make sine and cosine positive in every quadrant.

Ambiguous sine-rule case: if $a=8$, $b=10$ and $A=30^\circ$, then $\sin B=10\sin30^\circ/8=0.625$. Thus $B\approx38.68^\circ$ or $180^\circ-38.68^\circ=141.32^\circ$; both are valid because each gives $A+B<180^\circ$. Always test the supplementary angle against the triangle angle sum instead of accepting only the calculator's principal value.