Mathematical tables: trigonometry identities formulas
Tables show common trigonometric identities and formulas such as Pythagorean trigonometric identity, sine of half angle formula, etc.

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# Basic trigonometry identities#

 Name Formula Legend Tangent definition using sine and cosine functions Show source$tan(\alpha) = \frac{sin(\alpha)}{cos(\alpha)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function. Cotangent definition using cosine and sinefunctions Show source$cot(\alpha) = \frac{cos(\alpha)}{sin(\alpha)}$ $\alpha$ - the value of angle,cot - the cotangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent as cotangent inverse Show source$tan(\alpha) = \frac{1}{cot(\alpha)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cot - the cotangent of the angle function. Cotangent as tangent inverse Show source$cot(\alpha) = \frac{1}{tan(\alpha)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cot - the cotangent of the angle function. Pythagorean trigonometric identity Show source$sin^2(\alpha) + cos^2(\alpha) = 1$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Multiplication of tangent and cotangent of the same angle Show source$tan(\alpha) \cdot cot(\alpha) = 1$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cot - the cotangent of the angle function.

# Trigonometry: double-angle identities#

 Name Formula Legend Sine of double angle Show source$sin(2 \alpha) = 2 sin(\alpha) \cdot cos(\alpha)$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Cosine of double angle Show source$cos(2 \alpha) = cos^2(\alpha) - sin^2(\alpha) = 2 cos^2(\alpha) - 1$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent of double angle Show source$tan(2 \alpha) = \frac{2 tan(\alpha)}{1 - tan^2(\alpha)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function. Cotangent of double angle Show source$cot(2 \alpha) = \frac{cot^2(\alpha) - 1}{2 cot(\alpha)}$ $\alpha$ - the value of angle,cot - the cotangent of the angle function.

# Half-angle identities#

 Name Formula Legend Sine of half-angle Show source$sin\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - cos(\alpha)}{2}}$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Cosine of half-angle Show source$cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + cos(\alpha)}{2}}$ $\alpha$ - the value of angle,cos - the cosine of the angle function. Tangent of half-angle Show source$tan\left(\frac{\alpha}{2}\right) = \frac{1 - cos(\alpha)}{sin(\alpha)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function. Cotangent of half-angle Show source$cot\left(\frac{\alpha}{2}\right) = \frac{1 + cos(\alpha)}{sin(\alpha)}$ $\alpha$ - the value of angle,cot - the cotangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function.

# Angle-sum identities#

 Name Formula Legend Sine of angles sum Show source$sin(\alpha + \beta) = sin(\alpha) cos(\beta) + cos(\alpha) sin(\beta)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Cosine of angles sum Show source$cos(\alpha + \beta) = cos(\alpha) cos(\beta) - sin(\alpha) sin(\beta)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent of angles sum Show source$tan(\alpha + \beta) = \frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,tan - the tangent of the angle function. Cotangent of angles sum Show source$cot(\alpha + \beta) = \frac{cot(\alpha) \cdot cot(\beta) - 1}{cot(\alpha) + cot(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,cot - the cotangent of the angle function.

# Angle-difference identities#

 Name Formula Legend Sine of angles difference Show source$sin(\alpha - \beta) = sin(\alpha) cos(\beta) - cos(\alpha) sin(\beta)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Cosine of angles difference Show source$cos(\alpha - \beta) = cos(\alpha) cos(\beta) + sin(\alpha) sin(\beta)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent of angles difference Show source$tan(\alpha - \beta) = \frac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,tan - the tangent of the angle function. Cotangent of angles difference Show source$cot(\alpha - \beta) = \frac{cot(\alpha) \cdot cot(\beta) + 1}{cot(\alpha) - cot(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,cot - the cotangent of the angle function.

# Sum identities#

 Name Formula Legend Sum of sines Show source$sin(\alpha) + sin(\beta) = 2 sin\left(\frac{\alpha + \beta}{2}\right) \cdot cos\left(\frac{\alpha - \beta}{2}\right)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Sum of cosines Show source$cos(\alpha) + cos(\beta) = 2 cos\left(\frac{\alpha + \beta}{2}\right) \cdot cos\left(\frac{\alpha - \beta}{2}\right)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,cos - the cosine of the angle function. Sum of tangents Show source$tan(\alpha) + tan(\beta) = \frac{sin(\alpha + \beta)}{cos(\alpha) \cdot cos(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,tan - the tangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function. Sum of tangents Show source$cot(\alpha) + cot(\beta) = \frac{sin(\alpha + \beta)}{sin(\alpha) \cdot sin(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,cot - the cotangent of the angle function,sin - the sine of the angle function.

# Difference identities#

 Name Formula Legend Difference of sines Show source$sin(\alpha) - sin(\beta) = 2 sin\left(\frac{\alpha - \beta}{2}\right) \cdot cos\left(\frac{\alpha + \beta}{2}\right)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Difference of cosines Show source$cos(\alpha) - cos(\beta) = -2 sin\left(\frac{\alpha + \beta}{2}\right) \cdot sin\left(\frac{\alpha - \beta}{2}\right)$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,sin - the sine of the angle function,cos - the cosine of the angle function. Difference of tangents Show source$tan(\alpha) - tan(\beta) = \frac{sin(\alpha - \beta)}{cos(\alpha) \cdot cos(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,tan - the tangent of the angle function,sin - the sine of the angle function,cos - the cosine of the angle function. Difference of tangents Show source$cot(\alpha) - cot(\beta) = \frac{sin(\alpha - \beta)}{sin(\alpha) \cdot sin(\beta)}$ $\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,cot - the cotangent of the angle function,sin - the sine of the angle function.

# Periodicity of the trigonometric functions#

 Name Formula Legend Periodicity of the sine function Show source$sin(\alpha) = sin(\alpha + 2 k \pi)$ $\alpha$ - the value of angle,sin - the sine of the angle function. Periodicity of the cosine function Show source$cos(\alpha) = cos(\alpha + 2 k \pi)$ $\alpha$ - the value of angle,cos - the cosine of the angle function. Periodicity of the tangent function Show source$tan(\alpha) = tan(\alpha + k \pi)$ $\alpha$ - the value of angle,tan - the tangent of the angle function. Periodicity of the cotangent function Show source$cot(\alpha) = cot(\alpha + k \pi)$ $\alpha$ - the value of angle,cot - the cotangent of the angle function.

# Trigonometric functions in cosine representation#

 Name Formula Legend Sine in cosine form Show source$\left|sin(\alpha)\right| = \sqrt{1 - cos^2(\alpha)}$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent in cosine form Show source$\left|tan(\alpha)\right| = \frac{\sqrt{1 - cos^2(\alpha)}}{|cos(\alpha)|}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cos - the cosine of the angle function. Cotangent in cosine form Show source$\left|cot(\alpha)\right| = \frac{|cos(\alpha)|}{\sqrt{1 - cos^2(\alpha)}}$ $\alpha$ - the value of angle,cot - the cotangent of the angle function,cos - the cosine of the angle function.

# Trigonometric functions in sine representation#

 Name Formula Legend Cosine in sine form Show source$\left|cos(\alpha)\right| = \sqrt{1 - sin^2(\alpha)}$ $\alpha$ - the value of angle,sin - the sine of the angle function. Tangent in sine form Show source$\left|tan(\alpha)\right| = \frac{|sin(\alpha)|}{\sqrt{1 - sin^2(\alpha)}}$ $\alpha$ - the value of angle,tan - the tangent of the angle function,sin - the sine of the angle function. Cotangent in sine form Show source$\left|cot(\alpha)\right| = \frac{\sqrt{1 - sin^2(\alpha)}}{|sin(\alpha)|}$ $\alpha$ - the value of angle,cot - the cotangent of the angle function,sin - the sine of the angle function.

# Trigonometric functions in tangent of half-angle representation#

 Name Formula Legend Sine in tangent of half angle form Show source$sin(\alpha) = \frac{2 tan\left(\frac{\alpha}{2}\right)}{1 + tan^2\left(\frac{\alpha}{2}\right)}$ $\alpha$ - the value of angle,sin - the sine of the angle function,tan - the tangent of the angle function. Cosine in tangent of half angle form Show source$cos(\alpha) = \frac{1 - tan^2\left(\frac{\alpha}{2}\right)}{1 + tan^2\left(\frac{\alpha}{2}\right)}$ $\alpha$ - the value of angle,cos - the cosine of the angle function,tan - the tangent of the angle function. Tangent in tangent of half angle form Show source$tan(\alpha) = \frac{2 tan\left(\frac{\alpha}{2}\right)}{1 - tan^2\left(\frac{\alpha}{2}\right)}$ $\alpha$ - the value of angle,tan - the tangent of the angle function.

# Co-functions to functions relation#

 Name Formula Legend Sine to cosine relation Show source$sin(\alpha) = cos\left(\frac{\pi}{2} - \alpha\right)$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Cosine to sine relation Show source$cos(\alpha) = sin\left(\frac{\pi}{2} - \alpha\right)$ $\alpha$ - the value of angle,sin - the sine of the angle function,cos - the cosine of the angle function. Tangent to cotangent relation Show source$tan(\alpha) = cot\left(\frac{\pi}{2} - \alpha\right)$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cot - the cotangent of the angle function. Cotangent to tangent relation Show source$cot(\alpha) = tan\left(\frac{\pi}{2} - \alpha\right)$ $\alpha$ - the value of angle,tan - the tangent of the angle function,cot - the cotangent of the angle function.

# Parity of trigonometry functions#

 Name Formula Legend Sine of negative angle (odd function) Show source$sin(-\alpha) = -sin(\alpha)$ $\alpha$ - the value of angle,sin - the sine of the angle function. Cosine of negative angle (even function) Show source$cos(-\alpha) = cos(\alpha)$ $\alpha$ - the value of angle,cos - the cosine of the angle function. Tangent of negative angle (odd function) Show source$tan(-\alpha) = -tan(\alpha)$ $\alpha$ - the value of angle,tan - the tangent of the angle function. Cotangent of negative angle (odd function) Show source$cot(-\alpha) = -cot(\alpha)$ $\alpha$ - the value of angle,cot - the cotangent of the angle function.

# Euler's formulas#

 Name Formula Legend Euler formula Show source$e^{i x} = cos(x) + i \cdot sin(x)$ sin - the sine of the angle function,cos - the cosine of the angle function,e - e number (the base of natural logarithm),i - imaginary unit (complex number, root from -1). Sine as complex number form Show source$sin(x) = \frac{e^{i x} - e^{-i x}}{2i}$ sin - the sine of the angle function,e - e number (the base of natural logarithm),i - imaginary unit (complex number, root from -1). Cosine as complex number form Show source$cos(x) = \frac{e^{i x} + e^{-i x}}{2}$ cos - the cosine of the angle function,e - e number (the base of natural logarithm),i - imaginary unit (complex number, root from -1). Tangent as complex number form Show source$tan(x) = \frac{e^{i x} - e^{-i x}}{\left(e^{i x} + e^{-i x}\right) i}$ tan - the tangent of the angle function,e - e number (the base of natural logarithm),i - imaginary unit (complex number, root from -1). Cotangent as complex number form Show source$cot(x) = \frac{e^{i x} + e^{-i x}}{e^{i x} - e^{-i x}} i$ cot - the cotangent of the angle function,e - e number (the base of natural logarithm),i - imaginary unit (complex number, root from -1).

# De Moivre's formula#

 Name Formula Legend De Moivre's formula Show source$cos(n x) + i \cdot sin(n x) = \left(cos(x) + i \cdot sin(x)\right)^n$ sin - the sine of the angle function,cos - the cosine of the angle function,i - imaginary unit (complex number, root from -1). De Moivre's formula (general) Show source$\left[r(cos(x) + i \cdot sin(x)\right]^n = r^n\left(cos(n x) + i \cdot sin(n x)\right)$ sin - the sine of the angle function,cos - the cosine of the angle function,i - imaginary unit (complex number, root from -1).

# Product identities#

 Name Formula Legend Product of two cosine functions Show source$cos(\alpha) \cdot cos(\beta) = \frac{cos(\alpha - \beta) + cos(\alpha + \beta)}{2}$ cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle. The product of two sine functions Show source$sin(\alpha) \cdot sin(\beta) = \frac{cos(\alpha - \beta) - cos(\alpha + \beta)}{2}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle. Sine and cosine product Show source$sin(\alpha) \cdot cos(\beta) = \frac{sin(\alpha - \beta) + sin(\alpha + \beta)}{2}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle. Product of three sine functions Show source$sin(\alpha) \cdot sin(\beta) \cdot sin(\gamma) = \frac{sin(\alpha + \beta - \gamma) + sin(\beta + \gamma - \alpha) + sin(\gamma + \alpha - \beta) - sin(\alpha + \beta + \gamma)}{4}$ sin - the sine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,$\gamma$ - the value of the third angle. Product of two sine and one cosine functions Show source$sin(\alpha) \cdot sin(\beta) \cdot cos(\gamma) = \frac{-cos(\alpha + \beta - \gamma) + cos(\beta + \gamma - \alpha) + cos(\gamma + \alpha - \beta) - cos(\alpha + \beta + \gamma)}{4}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,$\gamma$ - the value of the third angle. Product of sine and two cosine functions Show source$sin(\alpha) \cdot cos(\beta) \cdot cos(\gamma) = \frac{sin(\alpha + \beta - \gamma) - sin(\beta + \gamma - \alpha) + sin(\gamma + \alpha - \beta) + sin(\alpha + \beta + \gamma)}{4}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,$\gamma$ - the value of the third angle. Product of three cosine functions Show source$cos(\alpha) \cdot cos(\beta) \cdot cos(\gamma) = \frac{cos(\alpha + \beta - \gamma) + cos(\beta + \gamma - \alpha) + cos(\gamma + \alpha - \beta) + cos(\alpha + \beta + \gamma)}{4}$ cos - the cosine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle,$\gamma$ - the value of the third angle.

# Power identities#

 Name Formula Legend Sine squared Show source$sin^2(\alpha) = \frac{1 - cos(2 \alpha)}{2}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of angle. Cosine squared Show source$cos^2(\alpha) = \frac{1 + cos(2 \alpha)}{2}$ cos - the cosine of the angle function,$\alpha$ - the value of angle. Sine squared times cosine squared Show source$sin^2(\alpha) \cdot cos^2(\alpha) = \frac{1 - cos(4 \alpha)}{8} = \frac{sin^2(2 \alpha)}{4}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of angle. Sine cubed Show source$sin^3(\alpha) = \frac{3 sin(\alpha) - sin(3 \alpha)}{4}$ sin - the sine of the angle function,$\alpha$ - the value of angle. Cosine cubed Show source$cos^3(\alpha) = \frac{3 cos(\alpha) + cos(3 \alpha)}{4}$ cos - the cosine of the angle function,$\alpha$ - the value of angle. Sine to the third power Show source$sin^4(\alpha) = \frac{cos(4 \alpha) - 4 cos(2 \alpha) + 3}{8}$ sin - the sine of the angle function,cos - the cosine of the angle function,$\alpha$ - the value of angle. Cosine to the third power Show source$cos^4(\alpha) = \frac{cos(4 \alpha) + 4 cos(2 \alpha) + 3}{8}$ cos - the cosine of the angle function,$\alpha$ - the value of angle. Sinus squared difference Show source$sin^2(\alpha) - sin^2(\beta) = sin(\alpha + \beta) \cdot sin(\alpha - \beta)$ sin - the sine of the angle function,$\alpha$ - the value of the first angle,$\beta$ - the value of the second angle.

# Some facts#

• Trigonometric identities are different dependencies between various trigonometric functions.
• Therefore, it is not a strict concept, and whether the expression is included in the trigonometric identity or not is purely practical.
• Typical trigonometric identities include:
• Pythagorean trigonometric identity:
$sin^2(\alpha) + cos^2(\alpha) = 1$
• the expressions to present one trigonometric function using another one e.g. the representation of tangent as the ratio of sine and cosine:
$tan(\alpha) = \frac{sin(\alpha)}{cos(\alpha)}$
• expressions for values of trigonometric functions for half-angles or multiples of angle, e.g.
$sin(2 \alpha) = 2 sin(\alpha) \cdot cos(\alpha)$
• expressions for sum, difference or product of one or more trigonometric functions, e.g.
$sin(\alpha) + sin(\beta) = 2 sin\left(\frac{\alpha + \beta}{2}\right) \cdot cos\left(\frac{\alpha - \beta}{2}\right)$
• expressions for values of the trigonometric function of angles sum or angles difference e.g.:
$sin(\alpha + \beta) = sin(\alpha) cos(\beta) + cos(\alpha) sin(\beta)$
• etc.
• ⓘ Hint: If you are interested in trigonometry you can checkout our other calculators:
• reduction formulas - so-called reduction formulas table, that help to calculate value of trigonometric functions for less common angles,
• trigonometric functions values - a table containing the values of trigonometric functions for the most common angles, e.g. sin 90 degrees,
• trigonometric identities - a list of different, more or less popular, dependencies between various trigonometric functions.