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.

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