Math · 37 · Geometry and Trigonometry

Trigonometry

Notes

SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
Opposite and adjacent depend on which angle you stand at.
Complementary angles: $\sin x^\circ=\cos(90^\circ-x^\circ)$ .
Ratios depend only on the angle — similar right triangles share them.
Radians: $\pi\text{ rad}=180^\circ$ , so $\tfrac{\pi}{6}=30^\circ$ , $\tfrac{\pi}{4}=45^\circ$ , $\tfrac{\pi}{3}=60^\circ$ .
Unit circle: the point at angle $\theta$ is $(\cos\theta,\ \sin\theta)$ .
Signs: $\sin>0$ in QI and QII; $\cos>0$ in QI and QIV.

Sine

\sin\theta=\dfrac{\text{opp}}{\text{hyp}}

Cosine

\cos\theta=\dfrac{\text{adj}}{\text{hyp}}

Tangent

\tan\theta=\dfrac{\text{opp}}{\text{adj}}=\dfrac{\sin\theta}{\cos\theta}

Cofunction

\sin\theta=\cos(90^\circ-\theta)

Not on the reference sheet

Degrees to radians

\text{radians}=\text{degrees}\times\dfrac{\pi}{180}

Special values

\sin30^\circ=\tfrac{1}{2},\ \ \cos30^\circ=\tfrac{\sqrt{3}}{2},\ \ \sin45^\circ=\cos45^\circ=\tfrac{\sqrt{2}}{2}

From the special right triangles

Pythagorean identity

\sin^2\theta+\cos^2\theta=1

Labelling sides from the wrong angle — opposite is across from the angle in use.
Desmos defaults to radians; switch to degrees before typing sin(30).
Reading $\sin x=\cos y$ as $x=y$ ; for acute angles, $x+y=90^\circ$ .

Given sin of one acute angle, the cosine of the other acute angle is the same number — no work needed.

10 official SAT questions on Trigonometry, easiest first.