Trigonometry

Trigonometry Formulas & Identities Cheat Sheet

Basic Trig Ratios

FunctionDefinitionMnemonic
sin⁡θ\sin\thetaOppositeHypotenuse\dfrac{\text{Opposite}}{\text{Hypotenuse}}SOH
cos⁡θ\cos\thetaAdjacentHypotenuse\dfrac{\text{Adjacent}}{\text{Hypotenuse}}CAH
tan⁡θ\tan\thetaOppositeAdjacent\dfrac{\text{Opposite}}{\text{Adjacent}}TOA
csc⁡θ\csc\thetaHypotenuseOpposite=1sin⁡θ\dfrac{\text{Hypotenuse}}{\text{Opposite}} = \dfrac{1}{\sin\theta}Reciprocal of sin
sec⁡θ\sec\thetaHypotenuseAdjacent=1cos⁡θ\dfrac{\text{Hypotenuse}}{\text{Adjacent}} = \dfrac{1}{\cos\theta}Reciprocal of cos
cot⁡θ\cot\thetaAdjacentOpposite=1tan⁡θ\dfrac{\text{Adjacent}}{\text{Opposite}} = \dfrac{1}{\tan\theta}Reciprocal of tan

Pythagorean Identities

IdentityDerived FormDerived Form
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1sin⁡2θ=1−cos⁡2θ\sin^2\theta = 1 - \cos^2\thetacos⁡2θ=1−sin⁡2θ\cos^2\theta = 1 - \sin^2\theta
tan⁡2θ+1=sec⁡2θ\tan^2\theta + 1 = \sec^2\thetatan⁡2θ=sec⁡2θ−1\tan^2\theta = \sec^2\theta - 1sec⁡2θ−tan⁡2θ=1\sec^2\theta - \tan^2\theta = 1
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\thetacot⁡2θ=csc⁡2θ−1\cot^2\theta = \csc^2\theta - 1csc⁡2θ−cot⁡2θ=1\csc^2\theta - \cot^2\theta = 1

Reciprocal Identities

IdentityEquivalent
csc⁡θ=1sin⁡θ\csc\theta = \dfrac{1}{\sin\theta}sin⁡θ=1csc⁡θ\sin\theta = \dfrac{1}{\csc\theta}
sec⁡θ=1cos⁡θ\sec\theta = \dfrac{1}{\cos\theta}cos⁡θ=1sec⁡θ\cos\theta = \dfrac{1}{\sec\theta}
cot⁡θ=1tan⁡θ\cot\theta = \dfrac{1}{\tan\theta}tan⁡θ=1cot⁡θ\tan\theta = \dfrac{1}{\cot\theta}

Quotient Identities

Identity
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}
cot⁡θ=cos⁡θsin⁡θ\cot\theta = \dfrac{\cos\theta}{\sin\theta}

Co-function Identities

IdentityIdentity
sin⁡θ=cos⁡(90°−θ)\sin\theta = \cos(90° - \theta)cos⁡θ=sin⁡(90°−θ)\cos\theta = \sin(90° - \theta)
tan⁡θ=cot⁡(90°−θ)\tan\theta = \cot(90° - \theta)cot⁡θ=tan⁡(90°−θ)\cot\theta = \tan(90° - \theta)
sec⁡θ=csc⁡(90°−θ)\sec\theta = \csc(90° - \theta)csc⁡θ=sec⁡(90°−θ)\csc\theta = \sec(90° - \theta)

Even/Odd Identities

FunctionIdentityType
sin⁡(−θ)\sin(-\theta)=−sin⁡θ= -\sin\thetaOdd
cos⁡(−θ)\cos(-\theta)=cos⁡θ= \cos\thetaEven
tan⁡(−θ)\tan(-\theta)=−tan⁡θ= -\tan\thetaOdd
csc⁡(−θ)\csc(-\theta)=−csc⁡θ= -\csc\thetaOdd
sec⁡(−θ)\sec(-\theta)=sec⁡θ= \sec\thetaEven
cot⁡(−θ)\cot(-\theta)=−cot⁡θ= -\cot\thetaOdd

Sum and Difference Formulas

Formula
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A\cos B + \cos A\sin B
sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A\cos B - \cos A\sin B
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A\cos B - \sin A\sin B
cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A\cos B + \sin A\sin B
tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A\tan B}
tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A\tan B}

Double-Angle Formulas

FormulaAlternate Forms
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta=2cos⁡2θ−1=1−2sin⁡2θ= 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \dfrac{2\tan\theta}{1 - \tan^2\theta}

Half-Angle Formulas

Formula
sin⁡ ⁣(θ2)=±1−cos⁡θ2\sin\!\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}
cos⁡ ⁣(θ2)=±1+cos⁡θ2\cos\!\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}
tan⁡ ⁣(θ2)=±1−cos⁡θ1+cos⁡θ=sin⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ\tan\!\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1 - \cos\theta}{1 + \cos\theta}} = \dfrac{\sin\theta}{1 + \cos\theta} = \dfrac{1 - \cos\theta}{\sin\theta}

The sign of the half-angle formulas depends on the quadrant of θ2\dfrac{\theta}{2}.

Power-Reducing Formulas

Formula
sin⁡2θ=1−cos⁡(2θ)2\sin^2\theta = \dfrac{1 - \cos(2\theta)}{2}
cos⁡2θ=1+cos⁡(2θ)2\cos^2\theta = \dfrac{1 + \cos(2\theta)}{2}
tan⁡2θ=1−cos⁡(2θ)1+cos⁡(2θ)\tan^2\theta = \dfrac{1 - \cos(2\theta)}{1 + \cos(2\theta)}

Product-to-Sum Formulas

Formula
sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \dfrac{1}{2}[\sin(A + B) + \sin(A - B)]
cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A\cos B = \dfrac{1}{2}[\cos(A - B) + \cos(A + B)]
sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A\sin B = \dfrac{1}{2}[\cos(A - B) - \cos(A + B)]

Law of Sines and Cosines

FormulaUse
asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}Two angles and a side, or two sides and an angle opposite one
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos CTwo sides and included angle, or three sides
Area=12absin⁡C\text{Area} = \dfrac{1}{2}ab\sin CTwo sides and included angle

Special Angle Values

Anglesin⁡\sincos⁡\costan⁡\tan
0°0°001100
30°30°12\dfrac{1}{2}32\dfrac{\sqrt{3}}{2}13=33\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}
45°45°22\dfrac{\sqrt{2}}{2}22\dfrac{\sqrt{2}}{2}11
60°60°32\dfrac{\sqrt{3}}{2}12\dfrac{1}{2}3\sqrt{3}
90°90°1100undefined

Degree-Radian Conversions

DegreesRadians
0°0°00
30°30°π6\dfrac{\pi}{6}
45°45°π4\dfrac{\pi}{4}
60°60°π3\dfrac{\pi}{3}
90°90°π2\dfrac{\pi}{2}
120°120°2π3\dfrac{2\pi}{3}
135°135°3π4\dfrac{3\pi}{4}
150°150°5π6\dfrac{5\pi}{6}
180°180°π\pi
270°270°3π2\dfrac{3\pi}{2}
360°360°2π2\pi

Conversion Formulas

ConversionFormula
Degrees to radiansradians=degrees×π180\text{radians} = \text{degrees} \times \dfrac{\pi}{180}
Radians to degreesdegrees=radians×180π\text{degrees} = \text{radians} \times \dfrac{180}{\pi}

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