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Trigonometry

Trigonometric ratios, identities, laws and angle formulae

Sine Ratio (SOH)

Basic
sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

Ratio of the side opposite an angle to the hypotenuse.

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Cosine Ratio (CAH)

Basic
cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

Ratio of the side adjacent to an angle to the hypotenuse.

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Tangent Ratio (TOA)

Basic
tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Ratio of the opposite side to the adjacent side.

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Cosecant

Intermediate
csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}

Reciprocal of sine.

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Secant

Intermediate
sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}

Reciprocal of cosine.

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Cotangent

Intermediate
cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Reciprocal of tangent.

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Tangent Identity

Basic
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}

Tangent equals sine divided by cosine.

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Pythagorean Identity

Basic
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

The fundamental trig identity from the unit circle.

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Identity: 1 + tan² = sec²

Intermediate
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

Derived Pythagorean identity dividing by cos².

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Identity: 1 + cot² = csc²

Intermediate
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Derived Pythagorean identity dividing by sin².

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Sine of a Sum

Intermediate
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A\cos B + \cos A\sin B

Angle addition formula for sine.

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Sine of a Difference

Intermediate
sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A-B) = \sin A\cos B - \cos A\sin B

Angle subtraction formula for sine.

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Cosine of a Sum

Intermediate
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A\cos B - \sin A\sin B

Angle addition formula for cosine.

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Cosine of a Difference

Intermediate
cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A-B) = \cos A\cos B + \sin A\sin B

Angle subtraction formula for cosine.

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Tangent of a Sum

Advanced
tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

Angle addition formula for tangent.

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Tangent of a Difference

Advanced
tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}

Angle subtraction formula for tangent.

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Double Angle: Sine

Intermediate
sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta

Sine of twice an angle.

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Double Angle: Cosine

Intermediate
cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta

Cosine of twice an angle (also 2cos²θ−1 or 1−2sin²θ).

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Double Angle: Tangent

Advanced
tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}

Tangent of twice an angle.

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Half Angle: Sine

Advanced
sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}

Sine of half an angle.

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Half Angle: Cosine

Advanced
cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}

Cosine of half an angle.

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Half Angle: Tangent

Advanced
tan⁡θ2=1−cos⁡θsin⁡θ\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Tangent of half an angle.

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Product: sin A cos B

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

Product-to-sum identity.

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Product: cos A cos B

Advanced
cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

Product-to-sum identity.

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Product: sin A sin B

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

Product-to-sum identity.

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Sum to Product: sin A + sin B

Advanced
sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

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Sum to Product: sin A − sin B

Advanced
sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

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Sum to Product: cos A + cos B

Advanced
cos⁡A+cos⁡B=2cos⁡A+B2cos⁡A−B2\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

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Sum to Product: cos A − cos B

Advanced
cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

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Law of Sines

Intermediate
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates sides of any triangle to the sines of opposite angles.

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Law of Cosines

Intermediate
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Generalises Pythagoras to any triangle.

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Law of Tangents

Advanced
a−ba+b=tan⁡A−B2tan⁡A+B2\frac{a-b}{a+b} = \frac{\tan\tfrac{A-B}{2}}{\tan\tfrac{A+B}{2}}

Relates two sides and their opposite angles.

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Triangle Area (SAS)

Intermediate
Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C

Area from two sides and the included angle.

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Heron's Formula

Intermediate
Area=s(s−a)(s−b)(s−c)\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from the three side lengths, where s is the semi-perimeter.

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Degrees to Radians

Basic
rad=deg×π180\text{rad} = \text{deg} \times \frac{\pi}{180}

Convert an angle from degrees to radians.

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Radians to Degrees

Basic
deg=rad×180π\text{deg} = \text{rad} \times \frac{180}{\pi}

Convert an angle from radians to degrees.

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Euler's Formula

Advanced
eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta

Links complex exponentials to trigonometry.

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De Moivre's Theorem

Advanced
(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

Powers of complex numbers in trig form.

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Sinusoidal Function

Intermediate
y=Asin⁡(Bx+C)+Dy = A\sin(Bx + C) + D

General sine wave with amplitude A, period 2π/B, phase C, shift D.

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