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Formulas: trigonometry, grade 11 (5 units)

Basic identities

  • The Pythagorean identity
    sin⁡2α+cos⁡2α\displaystyle \sin^{2}\alpha+\cos^{2}\alpha =\displaystyle = 1\displaystyle 1
  • The tangent
    tan⁡α\displaystyle \tan\alpha =\displaystyle = sin⁡αcos⁡α\displaystyle \frac{\sin\alpha}{\cos\alpha}cos⁡α≠0\displaystyle \cos\alpha\neq0
  • 1+tan⁡2α\displaystyle 1+\tan^{2}\alpha =\displaystyle = 1cos⁡2α\displaystyle \frac{1}{\cos^{2}\alpha}
  • sin⁡(−α)\displaystyle \sin(-\alpha) =\displaystyle = −sin⁡α\displaystyle -\sin\alpha
  • cos⁡(−α)\displaystyle \cos(-\alpha) =\displaystyle = cos⁡α\displaystyle \cos\alpha
  • sin⁡(π−α)\displaystyle \sin(\pi-\alpha) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • cos⁡(π−α)\displaystyle \cos(\pi-\alpha) =\displaystyle = −cos⁡α\displaystyle -\cos\alpha
  • sin⁡(π2−α)\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right) =\displaystyle = cos⁡α\displaystyle \cos\alpha
  • cos⁡(π2−α)\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right) =\displaystyle = sin⁡α\displaystyle \sin\alpha

The sum and the difference of angles

  • cos⁡(α±β)\displaystyle \cos(\alpha\pm\beta) =\displaystyle = cos⁡α⋅cos⁡β∓sin⁡α⋅sin⁡β\displaystyle \cos\alpha\cdot\cos\beta\mp\sin\alpha\cdot\sin\beta
  • sin⁡(α±β)\displaystyle \sin(\alpha\pm\beta) =\displaystyle = sin⁡α⋅cos⁡β±cos⁡α⋅sin⁡β\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta
  • tan⁡(α±β)\displaystyle \tan(\alpha\pm\beta) =\displaystyle = tan⁡α±tan⁡β1∓tan⁡α⋅tan⁡β\displaystyle \frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\cdot\tan\beta}

The double angle

  • sin⁡2α\displaystyle \sin2\alpha =\displaystyle = 2sin⁡α⋅cos⁡α\displaystyle 2\sin\alpha\cdot\cos\alpha
  • cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\displaystyle \cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha=2\cos^{2}\alpha-1=1-2\sin^{2}\alpha
  • tan⁡2α\displaystyle \tan2\alpha =\displaystyle = 2tan⁡α1−tan⁡2α\displaystyle \frac{2\tan\alpha}{1-\tan^{2}\alpha}
  • cos⁡2α\displaystyle \cos^{2}\alpha =\displaystyle = 1+cos⁡2α2\displaystyle \frac{1+\cos2\alpha}{2}
  • sin⁡2α\displaystyle \sin^{2}\alpha =\displaystyle = 1−cos⁡2α2\displaystyle \frac{1-\cos2\alpha}{2}

Sums and differences of sines and of cosines

  • sin⁡α+sin⁡β\displaystyle \sin\alpha+\sin\beta =\displaystyle = 2sin⁡α+β2⋅cos⁡α−β2\displaystyle 2\sin\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}
  • sin⁡α−sin⁡β\displaystyle \sin\alpha-\sin\beta =\displaystyle = 2sin⁡α−β2⋅cos⁡α+β2\displaystyle 2\sin\frac{\alpha-\beta}{2}\cdot\cos\frac{\alpha+\beta}{2}
  • cos⁡α+cos⁡β\displaystyle \cos\alpha+\cos\beta =\displaystyle = 2cos⁡α+β2⋅cos⁡α−β2\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}
  • cos⁡α−cos⁡β\displaystyle \cos\alpha-\cos\beta =\displaystyle = −2sin⁡α+β2⋅sin⁡α−β2\displaystyle -2\sin\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}

Equations – the general solution ($k$ a whole number)

  • sin⁡x=sin⁡α  ⟺  x=α+2πk or x=π−α+2πk\displaystyle \sin x=\sin\alpha\iff x=\alpha+2\pi k\ \text{or}\ x=\pi-\alpha+2\pi k
  • cos⁡x=cos⁡α  ⟺  x=±α+2πk\displaystyle \cos x=\cos\alpha\iff x=\pm\alpha+2\pi k
  • tan⁡x=tan⁡α  ⟺  x=α+πk\displaystyle \tan x=\tan\alpha\iff x=\alpha+\pi k

A triangle

  • The law of sines (R\displaystyle R – the radius of the circumscribed circle)
    asin⁡α=bsin⁡β=csin⁡γ=2R\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R
  • The law of cosines (γ\displaystyle \gamma – the angle between a\displaystyle a and b\displaystyle b)
    c2\displaystyle c^{2} =\displaystyle = a2+b2−2a⋅b⋅cos⁡γ\displaystyle a^{2}+b^{2}-2a\cdot b\cdot\cos\gamma
  • The area of a triangle (α\displaystyle \alpha – the angle between b\displaystyle b and c\displaystyle c)
    S\displaystyle S =\displaystyle = 12b⋅c⋅sin⁡α\displaystyle \frac{1}{2}b\cdot c\cdot\sin\alpha
  • The area of a triangle (β\displaystyle \beta, γ\displaystyle \gamma – the angles next to a\displaystyle a)
    S\displaystyle S =\displaystyle = a2⋅sin⁡β⋅sin⁡γ2⋅sin⁡(β+γ)\displaystyle \frac{a^{2}\cdot\sin\beta\cdot\sin\gamma}{2\cdot\sin(\beta+\gamma)}

A circle ($\alpha$ in radians, $R$ – the radius)

  • The area of a disc
    S\displaystyle S =\displaystyle = πR2\displaystyle \pi R^{2}
  • The circumference of a circle
    P\displaystyle P =\displaystyle = 2πR\displaystyle 2\pi R
  • The length of an arc
    l\displaystyle l =\displaystyle = α⋅R\displaystyle \alpha\cdot R
  • The area of a sector
    S\displaystyle S =\displaystyle = 12α⋅R2\displaystyle \frac{1}{2}\alpha\cdot R^{2}

Solids ($S$ – the base area, $h$ – the height, $l$ – the slant height)

  • The volume of a prism and a cylinder
    V\displaystyle V =\displaystyle = S⋅h\displaystyle S\cdot h
  • The volume of a pyramid and a cone
    V\displaystyle V =\displaystyle = S⋅h3\displaystyle \frac{S\cdot h}{3}
  • The lateral surface of a cone
    M\displaystyle M =\displaystyle = πR⋅l\displaystyle \pi R\cdot l
  • The volume of a sphere
    V\displaystyle V =\displaystyle = 43πR3\displaystyle \frac{4}{3}\pi R^{3}
  • The surface area of a sphere
    M\displaystyle M =\displaystyle = 4πR2\displaystyle 4\pi R^{2}