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

NameFormulaLegend
Tangent definition using sine and cosine functionsShow sourcetan(α)=sin(α)cos(α)tan(\alpha) = \frac{sin(\alpha)}{cos(\alpha)}
  • α\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 sinefunctionsShow sourcecot(α)=cos(α)sin(α)cot(\alpha) = \frac{cos(\alpha)}{sin(\alpha)}
  • α\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 inverseShow sourcetan(α)=1cot(α)tan(\alpha) = \frac{1}{cot(\alpha)}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cot - the cotangent of the angle function.
Cotangent as tangent inverseShow sourcecot(α)=1tan(α)cot(\alpha) = \frac{1}{tan(\alpha)}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cot - the cotangent of the angle function.
Pythagorean trigonometric identityShow sourcesin2(α)+cos2(α)=1sin^2(\alpha) + cos^2(\alpha) = 1
  • α\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 angleShow sourcetan(α)cot(α)=1tan(\alpha) \cdot cot(\alpha) = 1
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cot - the cotangent of the angle function.

Trigonometry: double-angle identities

NameFormulaLegend
Sine of double angleShow sourcesin(2α)=2sin(α)cos(α)sin(2 \alpha) = 2 sin(\alpha) \cdot cos(\alpha)
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Cosine of double angleShow sourcecos(2α)=cos2(α)sin2(α)=2cos2(α)1cos(2 \alpha) = cos^2(\alpha) - sin^2(\alpha) = 2 cos^2(\alpha) - 1
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Tangent of double angleShow sourcetan(2α)=2tan(α)1tan2(α)tan(2 \alpha) = \frac{2 tan(\alpha)}{1 - tan^2(\alpha)}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function.
Cotangent of double angleShow sourcecot(2α)=cot2(α)12cot(α)cot(2 \alpha) = \frac{cot^2(\alpha) - 1}{2 cot(\alpha)}
  • α\alpha - the value of angle,
  • cot - the cotangent of the angle function.

Half-angle identities

NameFormulaLegend
Sine of half-angleShow sourcesin(α2)=±1cos(α)2sin\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - cos(\alpha)}{2}}
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Cosine of half-angleShow sourcecos(α2)=±1+cos(α)2cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + cos(\alpha)}{2}}
  • α\alpha - the value of angle,
  • cos - the cosine of the angle function.
Tangent of half-angleShow sourcetan(α2)=1cos(α)sin(α)tan\left(\frac{\alpha}{2}\right) = \frac{1 - cos(\alpha)}{sin(\alpha)}
  • α\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-angleShow sourcecot(α2)=1+cos(α)sin(α)cot\left(\frac{\alpha}{2}\right) = \frac{1 + cos(\alpha)}{sin(\alpha)}
  • α\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

NameFormulaLegend
Sine of angles sumShow sourcesin(α+β)=sin(α)cos(β)+cos(α)sin(β)sin(\alpha + \beta) = sin(\alpha) cos(\beta) + cos(\alpha) sin(\beta)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Cosine of angles sumShow sourcecos(α+β)=cos(α)cos(β)sin(α)sin(β)cos(\alpha + \beta) = cos(\alpha) cos(\beta) - sin(\alpha) sin(\beta)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Tangent of angles sumShow sourcetan(α+β)=tan(α)+tan(β)1tan(α)tan(β)tan(\alpha + \beta) = \frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)}
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • tan - the tangent of the angle function.
Cotangent of angles sumShow sourcecot(α+β)=cot(α)cot(β)1cot(α)+cot(β)cot(\alpha + \beta) = \frac{cot(\alpha) \cdot cot(\beta) - 1}{cot(\alpha) + cot(\beta)}
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • cot - the cotangent of the angle function.

Angle-difference identities

NameFormulaLegend
Sine of angles differenceShow sourcesin(αβ)=sin(α)cos(β)cos(α)sin(β)sin(\alpha - \beta) = sin(\alpha) cos(\beta) - cos(\alpha) sin(\beta)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Cosine of angles differenceShow sourcecos(αβ)=cos(α)cos(β)+sin(α)sin(β)cos(\alpha - \beta) = cos(\alpha) cos(\beta) + sin(\alpha) sin(\beta)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Tangent of angles differenceShow sourcetan(αβ)=tan(α)tan(β)1+tan(α)tan(β)tan(\alpha - \beta) = \frac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)}
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • tan - the tangent of the angle function.
Cotangent of angles differenceShow sourcecot(αβ)=cot(α)cot(β)+1cot(α)cot(β)cot(\alpha - \beta) = \frac{cot(\alpha) \cdot cot(\beta) + 1}{cot(\alpha) - cot(\beta)}
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • cot - the cotangent of the angle function.

Sum identities

NameFormulaLegend
Sum of sinesShow sourcesin(α)+sin(β)=2sin(α+β2)cos(αβ2)sin(\alpha) + sin(\beta) = 2 sin\left(\frac{\alpha + \beta}{2}\right) \cdot cos\left(\frac{\alpha - \beta}{2}\right)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Sum of cosinesShow sourcecos(α)+cos(β)=2cos(α+β2)cos(αβ2)cos(\alpha) + cos(\beta) = 2 cos\left(\frac{\alpha + \beta}{2}\right) \cdot cos\left(\frac{\alpha - \beta}{2}\right)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • cos - the cosine of the angle function.
Sum of tangentsShow sourcetan(α)+tan(β)=sin(α+β)cos(α)cos(β)tan(\alpha) + tan(\beta) = \frac{sin(\alpha + \beta)}{cos(\alpha) \cdot cos(\beta)}
  • α\alpha - the value of the first angle,
  • β\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 tangentsShow sourcecot(α)+cot(β)=sin(α+β)sin(α)sin(β)cot(\alpha) + cot(\beta) = \frac{sin(\alpha + \beta)}{sin(\alpha) \cdot sin(\beta)}
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • cot - the cotangent of the angle function,
  • sin - the sine of the angle function.

Difference identities

NameFormulaLegend
Difference of sinesShow sourcesin(α)sin(β)=2sin(αβ2)cos(α+β2)sin(\alpha) - sin(\beta) = 2 sin\left(\frac{\alpha - \beta}{2}\right) \cdot cos\left(\frac{\alpha + \beta}{2}\right)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Difference of cosinesShow sourcecos(α)cos(β)=2sin(α+β2)sin(αβ2)cos(\alpha) - cos(\beta) = -2 sin\left(\frac{\alpha + \beta}{2}\right) \cdot sin\left(\frac{\alpha - \beta}{2}\right)
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Difference of tangentsShow sourcetan(α)tan(β)=sin(αβ)cos(α)cos(β)tan(\alpha) - tan(\beta) = \frac{sin(\alpha - \beta)}{cos(\alpha) \cdot cos(\beta)}
  • α\alpha - the value of the first angle,
  • β\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 tangentsShow sourcecot(α)cot(β)=sin(αβ)sin(α)sin(β)cot(\alpha) - cot(\beta) = \frac{sin(\alpha - \beta)}{sin(\alpha) \cdot sin(\beta)}
  • α\alpha - the value of the first angle,
  • β\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

NameFormulaLegend
Periodicity of the sine functionShow sourcesin(α)=sin(α+2kπ)sin(\alpha) = sin(\alpha + 2 k \pi)
  • α\alpha - the value of angle,
  • sin - the sine of the angle function.
Periodicity of the cosine functionShow sourcecos(α)=cos(α+2kπ)cos(\alpha) = cos(\alpha + 2 k \pi)
  • α\alpha - the value of angle,
  • cos - the cosine of the angle function.
Periodicity of the tangent functionShow sourcetan(α)=tan(α+kπ)tan(\alpha) = tan(\alpha + k \pi)
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function.
Periodicity of the cotangent functionShow sourcecot(α)=cot(α+kπ)cot(\alpha) = cot(\alpha + k \pi)
  • α\alpha - the value of angle,
  • cot - the cotangent of the angle function.

Trigonometric functions in cosine representation

NameFormulaLegend
Sine in cosine formShow sourcesin(α)=1cos2(α)\left|sin(\alpha)\right| = \sqrt{1 - cos^2(\alpha)}
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Tangent in cosine formShow sourcetan(α)=1cos2(α)cos(α)\left|tan(\alpha)\right| = \frac{\sqrt{1 - cos^2(\alpha)}}{|cos(\alpha)|}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cos - the cosine of the angle function.
Cotangent in cosine formShow sourcecot(α)=cos(α)1cos2(α)\left|cot(\alpha)\right| = \frac{|cos(\alpha)|}{\sqrt{1 - cos^2(\alpha)}}
  • α\alpha - the value of angle,
  • cot - the cotangent of the angle function,
  • cos - the cosine of the angle function.

Trigonometric functions in sine representation

NameFormulaLegend
Cosine in sine formShow sourcecos(α)=1sin2(α)\left|cos(\alpha)\right| = \sqrt{1 - sin^2(\alpha)}
  • α\alpha - the value of angle,
  • sin - the sine of the angle function.
Tangent in sine formShow sourcetan(α)=sin(α)1sin2(α)\left|tan(\alpha)\right| = \frac{|sin(\alpha)|}{\sqrt{1 - sin^2(\alpha)}}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • sin - the sine of the angle function.
Cotangent in sine formShow sourcecot(α)=1sin2(α)sin(α)\left|cot(\alpha)\right| = \frac{\sqrt{1 - sin^2(\alpha)}}{|sin(\alpha)|}
  • α\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

NameFormulaLegend
Sine in tangent of half angle formShow sourcesin(α)=2tan(α2)1+tan2(α2)sin(\alpha) = \frac{2 tan\left(\frac{\alpha}{2}\right)}{1 + tan^2\left(\frac{\alpha}{2}\right)}
  • α\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 formShow sourcecos(α)=1tan2(α2)1+tan2(α2)cos(\alpha) = \frac{1 - tan^2\left(\frac{\alpha}{2}\right)}{1 + tan^2\left(\frac{\alpha}{2}\right)}
  • α\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 formShow sourcetan(α)=2tan(α2)1tan2(α2)tan(\alpha) = \frac{2 tan\left(\frac{\alpha}{2}\right)}{1 - tan^2\left(\frac{\alpha}{2}\right)}
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function.

Co-functions to functions relation

NameFormulaLegend
Sine to cosine relationShow sourcesin(α)=cos(π2α)sin(\alpha) = cos\left(\frac{\pi}{2} - \alpha\right)
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Cosine to sine relationShow sourcecos(α)=sin(π2α)cos(\alpha) = sin\left(\frac{\pi}{2} - \alpha\right)
  • α\alpha - the value of angle,
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function.
Tangent to cotangent relationShow sourcetan(α)=cot(π2α)tan(\alpha) = cot\left(\frac{\pi}{2} - \alpha\right)
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cot - the cotangent of the angle function.
Cotangent to tangent relationShow sourcecot(α)=tan(π2α)cot(\alpha) = tan\left(\frac{\pi}{2} - \alpha\right)
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function,
  • cot - the cotangent of the angle function.

Parity of trigonometry functions

NameFormulaLegend
Sine of negative angle (odd function)Show sourcesin(α)=sin(α)sin(-\alpha) = -sin(\alpha)
  • α\alpha - the value of angle,
  • sin - the sine of the angle function.
Cosine of negative angle (even function)Show sourcecos(α)=cos(α)cos(-\alpha) = cos(\alpha)
  • α\alpha - the value of angle,
  • cos - the cosine of the angle function.
Tangent of negative angle (odd function)Show sourcetan(α)=tan(α)tan(-\alpha) = -tan(\alpha)
  • α\alpha - the value of angle,
  • tan - the tangent of the angle function.
Cotangent of negative angle (odd function)Show sourcecot(α)=cot(α)cot(-\alpha) = -cot(\alpha)
  • α\alpha - the value of angle,
  • cot - the cotangent of the angle function.

Euler's formulas

NameFormulaLegend
Euler formulaShow sourceeix=cos(x)+isin(x)e^{i x} = cos(x) + i \cdot sin(x)
Sine as complex number formShow sourcesin(x)=eixeix2isin(x) = \frac{e^{i x} - e^{-i x}}{2i}
Cosine as complex number formShow sourcecos(x)=eix+eix2cos(x) = \frac{e^{i x} + e^{-i x}}{2}
Tangent as complex number formShow sourcetan(x)=eixeix(eix+eix)itan(x) = \frac{e^{i x} - e^{-i x}}{\left(e^{i x} + e^{-i x}\right) i}
Cotangent as complex number formShow sourcecot(x)=eix+eixeixeixicot(x) = \frac{e^{i x} + e^{-i x}}{e^{i x} - e^{-i x}} i

De Moivre's formula

NameFormulaLegend
De Moivre's formulaShow sourcecos(nx)+isin(nx)=(cos(x)+isin(x))ncos(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[r(cos(x)+isin(x)]n=rn(cos(nx)+isin(nx))\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

NameFormulaLegend
Product of two cosine functionsShow sourcecos(α)cos(β)=cos(αβ)+cos(α+β)2cos(\alpha) \cdot cos(\beta) = \frac{cos(\alpha - \beta) + cos(\alpha + \beta)}{2}
  • cos - the cosine of the angle function,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle.
The product of two sine functionsShow sourcesin(α)sin(β)=cos(αβ)cos(α+β)2sin(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle.
Sine and cosine productShow sourcesin(α)cos(β)=sin(αβ)+sin(α+β)2sin(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle.
Product of three sine functionsShow sourcesin(α)sin(β)sin(γ)=sin(α+βγ)+sin(β+γα)+sin(γ+αβ)sin(α+β+γ)4sin(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • γ\gamma - the value of the third angle.
Product of two sine and one cosine functionsShow sourcesin(α)sin(β)cos(γ)=cos(α+βγ)+cos(β+γα)+cos(γ+αβ)cos(α+β+γ)4sin(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • γ\gamma - the value of the third angle.
Product of sine and two cosine functionsShow sourcesin(α)cos(β)cos(γ)=sin(α+βγ)sin(β+γα)+sin(γ+αβ)+sin(α+β+γ)4sin(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • γ\gamma - the value of the third angle.
Product of three cosine functionsShow sourcecos(α)cos(β)cos(γ)=cos(α+βγ)+cos(β+γα)+cos(γ+αβ)+cos(α+β+γ)4cos(\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,
  • α\alpha - the value of the first angle,
  • β\beta - the value of the second angle,
  • γ\gamma - the value of the third angle.

Power identities

NameFormulaLegend
Sine squaredShow sourcesin2(α)=1cos(2α)2sin^2(\alpha) = \frac{1 - cos(2 \alpha)}{2}
  • sin - the sine of the angle function,
  • cos - the cosine of the angle function,
  • α\alpha - the value of angle.
Cosine squaredShow sourcecos2(α)=1+cos(2α)2cos^2(\alpha) = \frac{1 + cos(2 \alpha)}{2}
  • cos - the cosine of the angle function,
  • α\alpha - the value of angle.
Sine squared times cosine squaredShow sourcesin2(α)cos2(α)=1cos(4α)8=sin2(2α)4sin^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,
  • α\alpha - the value of angle.
Sine cubedShow sourcesin3(α)=3sin(α)sin(3α)4sin^3(\alpha) = \frac{3 sin(\alpha) - sin(3 \alpha)}{4}
  • sin - the sine of the angle function,
  • α\alpha - the value of angle.
Cosine cubedShow sourcecos3(α)=3cos(α)+cos(3α)4cos^3(\alpha) = \frac{3 cos(\alpha) + cos(3 \alpha)}{4}
  • cos - the cosine of the angle function,
  • α\alpha - the value of angle.
Sine to the third powerShow sourcesin4(α)=cos(4α)4cos(2α)+38sin^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,
  • α\alpha - the value of angle.
Cosine to the third powerShow sourcecos4(α)=cos(4α)+4cos(2α)+38cos^4(\alpha) = \frac{cos(4 \alpha) + 4 cos(2 \alpha) + 3}{8}
  • cos - the cosine of the angle function,
  • α\alpha - the value of angle.
Sinus squared differenceShow sourcesin2(α)sin2(β)=sin(α+β)sin(αβ)sin^2(\alpha) - sin^2(\beta) = sin(\alpha + \beta) \cdot sin(\alpha - \beta)
  • sin - the sine of the angle function,
  • α\alpha - the value of the first angle,
  • β\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:
      sin2(α)+cos2(α)=1sin^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(α)=sin(α)cos(α)tan(\alpha) = \frac{sin(\alpha)}{cos(\alpha)}
    • expressions for values of trigonometric functions for half-angles or multiples of angle, e.g.
      sin(2α)=2sin(α)cos(α)sin(2 \alpha) = 2 sin(\alpha) \cdot cos(\alpha)
    • expressions for sum, difference or product of one or more trigonometric functions, e.g.
      sin(α)+sin(β)=2sin(α+β2)cos(αβ2)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(α+β)=sin(α)cos(β)+cos(α)sin(β)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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