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Trigonometry Formulas

Measurement & Conversion
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Even-Odd Identities
Cofunction Identities
Supplement & Shift Identities
Periodicity Identities
Sum & Difference Formulas
Double Angle Formulas
Half-Angle Formulas
Triple Angle Formulas
Power-Reducing Formulas
Product-to-Sum Formulas
Sum-to-Product Formulas
Triangle Formulas
Inverse Trigonometric Identities
General Solutions
62 formulas

Measurement & Conversion

(3 formulas)

Degree-Radian Conversion

θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}
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Converts an angle from degrees to radians by multiplying by the ratio π180\frac{\pi}{180}. The reverse conversion multiplies by 180π\frac{180}{\pi}.
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Arc Length

s=rθs = r\theta
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The length of a circular arc equals the radius times the central angle in radians. This is a direct consequence of the radian definition θ=sr\theta = \frac{s}{r}.
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Sector Area

A=12r2θA = \frac{1}{2}r^2\theta
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The area of a circular sector equals half the square of the radius times the central angle in radians. The sector is the fraction θ2π\frac{\theta}{2\pi} of the full circle area πr2\pi r^2.
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Reciprocal Identities

(3 formulas)

Cosecant Reciprocal Identity

csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}
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Cosecant is the reciprocal of sine. The product sin⁡θ⋅csc⁡θ=1\sin\theta \cdot \csc\theta = 1 holds wherever both are defined.
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Secant Reciprocal Identity

sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}
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explanationconditionsrelated formulasrelated definitions
Secant is the reciprocal of cosine. The product cos⁡θ⋅sec⁡θ=1\cos\theta \cdot \sec\theta = 1 holds wherever both are defined.
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Cotangent Reciprocal Identity

cot⁡θ=1tan⁡θ\cot\theta = \frac{1}{\tan\theta}
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Cotangent is the reciprocal of tangent. The product tan⁡θ⋅cot⁡θ=1\tan\theta \cdot \cot\theta = 1 holds wherever both are defined.
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Quotient Identities

(2 formulas)

Tangent Quotient Identity

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
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Tangent equals the ratio of sine to cosine. On the unit circle, this is yx\frac{y}{x}, which geometrically represents the slope of the terminal side.
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Cotangent Quotient Identity

cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}
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explanationconditionsrelated formulasrelated definitions
Cotangent equals the ratio of cosine to sine — the reciprocal of the tangent quotient. On the unit circle, this is xy\frac{x}{y}.
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Pythagorean Identities

(3 formulas)

Pythagorean Identity - Sine Cosine

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
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The fundamental trigonometric identity. It restates the unit circle equation x2+y2=1x^2 + y^2 = 1 with x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta.
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Pythagorean Identity - Tangent Secant

1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
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Obtained by dividing sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 by cos⁡2θ\cos^2\theta. Connects tangent and secant through the same Pythagorean structure.
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Pythagorean Identity - Cotangent Cosecant

1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta
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explanationderivationconditionsvariantsrelated formulasrelated definitions
Obtained by dividing sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 by sin⁡2θ\sin^2\theta. Connects cotangent and cosecant through the same Pythagorean structure.
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Even-Odd Identities

(3 formulas)

Sine Odd Identity

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta
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Sine is an odd function. Negating the angle negates the output. On the unit circle, reflecting across the xx-axis negates the yy-coordinate.
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Cosine Even Identity

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta
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Cosine is an even function. Negating the angle leaves the output unchanged. On the unit circle, reflecting across the xx-axis preserves the xx-coordinate.
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Tangent Odd Identity

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan\theta
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explanationvariantsrelated formulasrelated definitions
Tangent is an odd function. It is the ratio of sine (odd) to cosine (even), so −sin⁡θcos⁡θ=−tan⁡θ\frac{-\sin\theta}{\cos\theta} = -\tan\theta.
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Cofunction Identities

(3 formulas)

Sine-Cosine Cofunction

sin⁡ ⁣(π2−θ)=cos⁡θ\sin\!\left(\frac{\pi}{2} - \theta\right) = \cos\theta
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The sine of an angle equals the cosine of its complement. In a right triangle, the side opposite one acute angle is adjacent to the other.
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Cosine-Sine Cofunction

cos⁡ ⁣(π2−θ)=sin⁡θ\cos\!\left(\frac{\pi}{2} - \theta\right) = \sin\theta
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The cosine of an angle equals the sine of its complement. This is the symmetric counterpart of the sine cofunction identity.
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Tangent-Cotangent Cofunction

tan⁡ ⁣(π2−θ)=cot⁡θ\tan\!\left(\frac{\pi}{2} - \theta\right) = \cot\theta
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The tangent of an angle equals the cotangent of its complement. Follows from dividing the sine and cosine cofunction identities.
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Supplement & Shift Identities

(6 formulas)

Sine Supplement Identity

sin⁡(π−θ)=sin⁡θ\sin(\pi - \theta) = \sin\theta
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Supplementary angles share the same sine. On the unit circle, the points at θ\theta and π−θ\pi - \theta are reflections across the yy-axis, preserving the yy-coordinate.
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Cosine Supplement Identity

cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos\theta
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Supplementary angles have opposite cosines. Reflecting across the yy-axis negates the xx-coordinate.
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Tangent Supplement Identity

tan⁡(π−θ)=−tan⁡θ\tan(\pi - \theta) = -\tan\theta
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The tangent of a supplementary angle is negated. Follows from sin⁡θ−cos⁡θ=−tan⁡θ\frac{\sin\theta}{-\cos\theta} = -\tan\theta.
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Sine Anti-Supplement Shift

sin⁡(π+θ)=−sin⁡θ\sin(\pi + \theta) = -\sin\theta
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Adding π\pi to the angle moves the point to the diametrically opposite position on the unit circle, negating both coordinates.
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Cosine Anti-Supplement Shift

cos⁡(π+θ)=−cos⁡θ\cos(\pi + \theta) = -\cos\theta
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Adding π\pi negates the xx-coordinate on the unit circle. Combined with sin⁡(π+θ)=−sin⁡θ\sin(\pi + \theta) = -\sin\theta, this implies tan⁡(π+θ)=tan⁡θ\tan(\pi + \theta) = \tan\theta (the basis for tangent's π\pi period).
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Quadrature Shift - Sine

sin⁡ ⁣(π2+θ)=cos⁡θ\sin\!\left(\frac{\pi}{2} + \theta\right) = \cos\theta
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Adding π2\frac{\pi}{2} to the argument of sine produces cosine. This is the forward-shift counterpart of the cofunction identity (which subtracts π2\frac{\pi}{2}).
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Periodicity Identities

(3 formulas)

Sine Periodicity

sin⁡(θ+2π)=sin⁡θ\sin(\theta + 2\pi) = \sin\theta
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Sine repeats every 2π2\pi radians. A full rotation returns the point on the unit circle to its starting position, so the yy-coordinate is unchanged.
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Cosine Periodicity

cos⁡(θ+2π)=cos⁡θ\cos(\theta + 2\pi) = \cos\theta
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Cosine repeats every 2π2\pi radians. A full rotation preserves the xx-coordinate on the unit circle.
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Tangent Periodicity

tan⁡(θ+π)=tan⁡θ\tan(\theta + \pi) = \tan\theta
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Tangent has the shorter period π\pi. A half rotation negates both coordinates, but the ratio yx\frac{y}{x} is unchanged.
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Sum & Difference Formulas

(3 formulas)

Sine Sum and Difference

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
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Expresses the sine of a sum or difference of two angles in terms of sines and cosines of the individual angles. The ±\pm signs match on both sides.
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Cosine Sum and Difference

cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
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Expresses the cosine of a sum or difference. Note the sign reversal: the ±\pm on the left corresponds to ∓\mp on the right.
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Tangent Sum and Difference

tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
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Expresses the tangent of a sum or difference. The numerator sign matches the left side; the denominator sign is opposite.
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Double Angle Formulas

(3 formulas)

Sine Double Angle

sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2\sin A\cos A
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The sine of twice an angle equals twice the product of its sine and cosine. Obtained by setting B=AB = A in the sine sum formula.
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Cosine Double Angle

cos⁡2A=cos⁡2A−sin⁡2A\cos 2A = \cos^2 A - \sin^2 A
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The cosine of twice an angle. This identity has three standard forms, obtained by applying sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 to eliminate one function.
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Tangent Double Angle

tan⁡2A=2tan⁡A1−tan⁡2A\tan 2A = \frac{2\tan A}{1 - \tan^2 A}
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The tangent of twice an angle. Obtained by setting B=AB = A in the tangent sum formula.
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Half-Angle Formulas

(3 formulas)

Sine Half Angle

sin⁡A2=±1−cos⁡A2\sin\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}
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Expresses the sine of half an angle in terms of the cosine of the full angle. The sign depends on the quadrant of A2\frac{A}{2}.
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Cosine Half Angle

cos⁡A2=±1+cos⁡A2\cos\frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}
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Expresses the cosine of half an angle in terms of the cosine of the full angle. The sign depends on the quadrant of A2\frac{A}{2}.
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Tangent Half Angle

tan⁡A2=±1−cos⁡A1+cos⁡A\tan\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}}
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Expresses the tangent of half an angle. The sign-free alternate forms avoid the ±\pm ambiguity entirely.
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Triple Angle Formulas

(3 formulas)

Sine Triple Angle

sin⁡3A=3sin⁡A−4sin⁡3A\sin 3A = 3\sin A - 4\sin^3 A
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Expresses sin⁡3A\sin 3A purely in terms of sin⁡A\sin A. Useful for reducing higher-multiple arguments to single-angle expressions.
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Cosine Triple Angle

cos⁡3A=4cos⁡3A−3cos⁡A\cos 3A = 4\cos^3 A - 3\cos A
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Expresses cos⁡3A\cos 3A purely in terms of cos⁡A\cos A. The Chebyshev polynomial T3(x)=4x3−3xT_3(x) = 4x^3 - 3x has the same form.
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Tangent Triple Angle

tan⁡3A=3tan⁡A−tan⁡3A1−3tan⁡2A\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}
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Expresses tan⁡3A\tan 3A in terms of tan⁡A\tan A.
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Power-Reducing Formulas

(3 formulas)

Sine Squared Reduction

sin⁡2A=1−cos⁡2A2\sin^2 A = \frac{1 - \cos 2A}{2}
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Replaces sin⁡2A\sin^2 A with a first-power expression involving cos⁡2A\cos 2A. Essential for integration of even powers of sine.
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Cosine Squared Reduction

cos⁡2A=1+cos⁡2A2\cos^2 A = \frac{1 + \cos 2A}{2}
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Replaces cos⁡2A\cos^2 A with a first-power expression involving cos⁡2A\cos 2A. Essential for integration of even powers of cosine.
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Tangent Squared Reduction

tan⁡2A=1−cos⁡2A1+cos⁡2A\tan^2 A = \frac{1 - \cos 2A}{1 + \cos 2A}
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Replaces tan⁡2A\tan^2 A with an expression involving cos⁡2A\cos 2A. Obtained by dividing the sine reduction by the cosine reduction.
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Product-to-Sum Formulas

(4 formulas)

Product to Sum - Cosine Cosine

cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A\cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]
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Converts a product of two cosines into a sum of cosines. Derived by adding the cosine sum and difference formulas.
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Product to Sum - Sine Sine

sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A\sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]
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Converts a product of two sines into a difference of cosines. Derived by subtracting the cosine sum formula from the cosine difference formula.
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Product to Sum - Sine Cosine

sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]
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Converts a product of sine and cosine into a sum of sines. Derived by adding the sine sum and difference formulas.
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Product to Sum - Cosine Sine

cos⁡Asin⁡B=12[sin⁡(A+B)−sin⁡(A−B)]\cos A\sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]
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Converts a product of cosine and sine into a difference of sines. The order of AA and BB matters — swapping them changes the sign pattern.
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Sum-to-Product Formulas

(4 formulas)

Sum to Product - Sine Sum

sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A + \sin B = 2\sin\frac{A + B}{2}\cos\frac{A - B}{2}
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Converts a sum of sines into a product of sine and cosine. The arguments are the half-sum and half-difference of the original angles.
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Sum to Product - Sine Difference

sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A - \sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}
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Converts a difference of sines into a product of cosine and sine. Note the switch: the sum formula has sine first, the difference has cosine first.
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Sum to Product - Cosine Sum

cos⁡A+cos⁡B=2cos⁡A+B2cos⁡A−B2\cos A + \cos B = 2\cos\frac{A + B}{2}\cos\frac{A - B}{2}
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Converts a sum of cosines into a product of two cosines evaluated at the half-sum and half-difference.
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Sum to Product - Cosine Difference

cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A - \cos B = -2\sin\frac{A + B}{2}\sin\frac{A - B}{2}
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Converts a difference of cosines into a product. Note the leading negative sign — this is the only sum-to-product formula with a minus in front.
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Triangle Formulas

(5 formulas)

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
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In any triangle, each side is proportional to the sine of its opposite angle. The common ratio equals the diameter of the circumscribed circle (2R2R).
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Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
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Relates the three sides of any triangle to the cosine of one angle. Generalizes the Pythagorean theorem: when C=90°C = 90°, the formula reduces to c2=a2+b2c^2 = a^2 + b^2.
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Law of Tangents

a−ba+b=tan⁡A−B2tan⁡A+B2\frac{a - b}{a + b} = \frac{\tan\frac{A - B}{2}}{\tan\frac{A + B}{2}}
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Relates the difference and sum of two sides to the tangent of the half-difference and half-sum of their opposite angles. Less commonly used than the laws of sines and cosines, but avoids the ambiguous case.
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Triangle Area - SAS

A=12absin⁡CA = \frac{1}{2}ab\sin C
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The area of a triangle equals half the product of two sides times the sine of the included angle. Works for all triangles — the standard base-times-height formula is the special case where C=90°C = 90°.
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Herons Formula

A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}
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Computes the area of a triangle from its three side lengths alone, with no angle information required. The semi-perimeter ss absorbs the side data into a symmetric form.
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Inverse Trigonometric Identities

(5 formulas)

Arcsin Plus Arccos

arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2}
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The arcsine and arccosine of the same value are complementary angles. This is the inverse-function version of the cofunction identity sin⁡θ=cos⁡ ⁣(π2−θ)\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right).
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Arctan Plus Arccot

arctan⁡x+arccot⁡x=π2\arctan x + \operatorname{arccot} x = \frac{\pi}{2}
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The arctangent and arccotangent of the same value are complementary. Parallels the arcsin-plus-arccos identity for the tangent-cotangent pair.
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Arcsin of Negative

arcsin⁡(−x)=−arcsin⁡x\arcsin(-x) = -\arcsin x
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Arcsine is an odd function, inheriting the odd symmetry of sine. Negating the input negates the output.
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Arccos of Negative

arccos⁡(−x)=π−arccos⁡x\arccos(-x) = \pi - \arccos x
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Arccosine is not odd — negating the input supplements the output rather than negating it. This follows from the supplement identity cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos\theta.
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Arctan of Negative

arctan⁡(−x)=−arctan⁡x\arctan(-x) = -\arctan x
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Arctangent is an odd function, inheriting the odd symmetry of tangent. Negating the input negates the output.
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General Solutions

(3 formulas)

General Solution - Sine Equation

sin⁡θ=k  ⟹  θ=(−1)narcsin⁡k+nπ\sin\theta = k \implies \theta = (-1)^n \arcsin k + n\pi
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Captures all solutions to sin⁡θ=k\sin\theta = k in a single expression. The alternating sign (−1)n(-1)^n accounts for sine having two solutions per 2π2\pi period — one in Quadrant I/II and one reflected.
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General Solution - Cosine Equation

cos⁡θ=k  ⟹  θ=±arccos⁡k+2nπ\cos\theta = k \implies \theta = \pm\arccos k + 2n\pi
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Captures all solutions to cos⁡θ=k\cos\theta = k. The ±\pm reflects cosine's even symmetry — both +arccos⁡k+\arccos k and −arccos⁡k-\arccos k are solutions.
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General Solution - Tangent Equation

tan⁡θ=k  ⟹  θ=arctan⁡k+nπ\tan\theta = k \implies \theta = \arctan k + n\pi
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Captures all solutions to tan⁡θ=k\tan\theta = k. The simplest general solution — tangent has period π\pi and one solution per period, so a single family suffices.
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Measurement & Conversion
Degree-Radian ConversionArc LengthSector Area
Reciprocal Identities
Cosecant Reciprocal IdentitySecant Reciprocal IdentityCotangent Reciprocal Identity
Quotient Identities
Tangent Quotient IdentityCotangent Quotient Identity
Pythagorean Identities
Pythagorean Identity - Sine CosinePythagorean Identity - Tangent SecantPythagorean Identity - Cotangent Cosecant
Even-Odd Identities
Sine Odd IdentityCosine Even IdentityTangent Odd Identity
Cofunction Identities
Sine-Cosine CofunctionCosine-Sine CofunctionTangent-Cotangent Cofunction
Supplement & Shift Identities
Sine Supplement IdentityCosine Supplement IdentityTangent Supplement IdentitySine Anti-Supplement ShiftCosine Anti-Supplement ShiftQuadrature Shift - Sine
Periodicity Identities
Sine PeriodicityCosine PeriodicityTangent Periodicity
Sum & Difference Formulas
Sine Sum and DifferenceCosine Sum and DifferenceTangent Sum and Difference
Double Angle Formulas
Sine Double AngleCosine Double AngleTangent Double Angle
Half-Angle Formulas
Sine Half AngleCosine Half AngleTangent Half Angle
Triple Angle Formulas
Sine Triple AngleCosine Triple AngleTangent Triple Angle
Power-Reducing Formulas
Sine Squared ReductionCosine Squared ReductionTangent Squared Reduction
Product-to-Sum Formulas
Product to Sum - Cosine CosineProduct to Sum - Sine SineProduct to Sum - Sine CosineProduct to Sum - Cosine Sine
Sum-to-Product Formulas
Sum to Product - Sine SumSum to Product - Sine DifferenceSum to Product - Cosine SumSum to Product - Cosine Difference
Triangle Formulas
Law of SinesLaw of CosinesLaw of TangentsTriangle Area - SASHerons Formula
Inverse Trigonometric Identities
Arcsin Plus ArccosArctan Plus ArccotArcsin of NegativeArccos of NegativeArctan of Negative
General Solutions
General Solution - Sine EquationGeneral Solution - Cosine EquationGeneral Solution - Tangent Equation