Trigonometry: SOH CAH TOA

Hypotenuse

Opposite

Adjacent

\dfrac{\text{opposite}}{\text{hypotenuse}}

\dfrac{\text{adjacent}}{\text{hypotenuse}}

\dfrac{\text{opposite}}{\text{adjacent}}

\tfrac{1}{2}

It follows from a² + b² = c² on the unit circle.

So 180° = π radians.

5

Both equal \dfrac{\sqrt{2}}{2}

\dfrac{\sqrt{3}}{2}

\sin 60^\circ = \dfrac{\sqrt{3}}{2} , \cos 60^\circ = \dfrac{1}{2}

\dfrac{\sqrt{3}}{3} , 1 and \sqrt{3}

1 : \sqrt{3} : 2 (short leg : long leg : hypotenuse)

1 : 1 : \sqrt{2}

\csc\theta = \dfrac{1}{\sin\theta} , \sec\theta = \dfrac{1}{\cos\theta} , \cot\theta = \dfrac{1}{\tan\theta}

\tan\theta = \dfrac{\sin\theta}{\cos\theta}

Degrees to radians: multiply by \dfrac{\pi}{180} Radians to degrees: multiply by \dfrac{180}{\pi}

\dfrac{\pi}{3}

135

(\cos\theta, \sin\theta)

QI: all QII: sine (and csc) QIII: tangent (and cot) QIV: cosine (and sec)

The acute angle between the terminal side of an angle and the x-axis. Trig values of an angle equal those of its reference angle, up to sign.

\tfrac{1}{2}

-\tfrac{1}{2}

1 , 0 , and undefined (division by \cos 90^\circ = 0 )

\sin and \cos : range [-1, 1] , period 2\pi \tan : range all real numbers, period \pi

\theta = \sin^{-1}\left(\dfrac{5}{10}\right) = 30^\circ

5

0.75

Elevation: measured up from the horizontal to an object above you. Depression: measured down from the horizontal to an object below you.

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

1 + \tan^2\theta = \sec^2\theta 1 + \cot^2\theta = \csc^2\theta

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

c^2 = a^2 + b^2 - 2ab\cos C

\text{Area} = \tfrac{1}{2}ab\sin C

3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples, like 6-8-10)