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Formulas: trigonometry, grade 10 (5 units)

The unit circle (radians)

  • The unit circle
    x2+y2=1\displaystyle x^2+y^2=1
  • The point of the number t\displaystyle t on the unit circle
    P(t)=(cos⁡t, sin⁡t)\displaystyle P(t)=(\cos t,\ \sin t)
  • All the numbers with the same point as t\displaystyle t (k\displaystyle k an integer)
    t+2πk\displaystyle t+2\pi k
  • Radians and degrees
    π\displaystyle \pi =\displaystyle = 180∘\displaystyle 180^{\circ}
  • The length of an arc of α\displaystyle \alpha radians (R\displaystyle R – the radius)
    ℓ\displaystyle \ell =\displaystyle = αR\displaystyle \alpha R
  • The area of a sector of α\displaystyle \alpha radians (R\displaystyle R – the radius)
    S\displaystyle S =\displaystyle = 12αR2\displaystyle \frac{1}{2}\alpha R^2

Special values

  • sin⁡π6=cos⁡π3=12\displaystyle \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}
  • sin⁡π4=cos⁡π4=22\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}
  • sin⁡π3=cos⁡π6=32\displaystyle \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}
  • tan⁡π6=33,tan⁡π4=1,tan⁡π3=3\displaystyle \tan\frac{\pi}{6}=\frac{\sqrt{3}}{3},\quad\tan\frac{\pi}{4}=1,\quad\tan\frac{\pi}{3}=\sqrt{3}

Identities

  • The basic identity
    sin⁡2α+cos⁡2α\displaystyle \sin^2\alpha+\cos^2\alpha =\displaystyle = 1\displaystyle 1
  • 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⁡(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
  • sin⁡(α+2πk)\displaystyle \sin(\alpha+2\pi k) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • tan⁡(α+πk)\displaystyle \tan(\alpha+\pi k) =\displaystyle = tan⁡α\displaystyle \tan\alpha

Graphs of trigonometric functions

  • A general sine (and a cosine likewise)
    y=Asin⁡(ax+b)+B\displaystyle y=A\sin(ax+b)+B
  • The period of sin⁡(ax+b)\displaystyle \sin(ax+b) and of cos⁡(ax+b)\displaystyle \cos(ax+b)
    T\displaystyle T =\displaystyle = 2π∣a∣\displaystyle \frac{2\pi}{|a|}
  • The range of Asin⁡(ax+b)+B\displaystyle A\sin(ax+b)+B
    B−∣A∣≤y≤B+∣A∣\displaystyle B-|A|\leq y\leq B+|A|
  • The period of tan⁡(ax+b)\displaystyle \tan(ax+b)
    T\displaystyle T =\displaystyle = π∣a∣\displaystyle \frac{\pi}{|a|}

Trigonometric equations ($k$ an integer)

  • The solutions of sin⁡x=sin⁡α\displaystyle {\sin x}={\sin\alpha}
    x=α+2πk,x=π−α+2πk\displaystyle x=\alpha+2\pi k,\quad x=\pi-\alpha+2\pi k
  • The solutions of cos⁡x=cos⁡α\displaystyle {\cos x}={\cos\alpha}
    x=±α+2πk\displaystyle x=\pm\alpha+2\pi k
  • The solutions of tan⁡x=tan⁡α\displaystyle {\tan x}={\tan\alpha}
    x=α+πk\displaystyle x=\alpha+\pi k

Trigonometry in a triangle (degrees)

  • In a right-angled triangle: a\displaystyle a the leg opposite α\displaystyle \alpha, b\displaystyle b the leg beside it, c\displaystyle c the hypotenuse
    sin⁡α=ac,cos⁡α=bc,tan⁡α=ab\displaystyle \sin\alpha=\frac{a}{c},\quad\cos\alpha=\frac{b}{c},\quad\tan\alpha=\frac{a}{b}
  • sin⁡(180∘−α)\displaystyle \sin(180^{\circ}-\alpha) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • cos⁡(180∘−α)\displaystyle \cos(180^{\circ}-\alpha) =\displaystyle = −cos⁡α\displaystyle -\cos\alpha
  • The area of a triangle (α\displaystyle \alpha – the angle between the sides b\displaystyle b and c\displaystyle c)
    S\displaystyle S =\displaystyle = 12bcsin⁡α\displaystyle \frac{1}{2}bc\sin\alpha
  • The area of a triangle (β\displaystyle \beta and γ\displaystyle \gamma – the angles at the side a\displaystyle a)
    S\displaystyle S =\displaystyle = a2sin⁡βsin⁡γ2sin⁡(β+γ)\displaystyle \frac{a^2\sin\beta\sin\gamma}{2\sin(\beta+\gamma)}
  • The law of sines (R\displaystyle R – the radius of the circle that circumscribes the triangle)
    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 the sides a\displaystyle a and b\displaystyle b)
    c2\displaystyle c^2 =\displaystyle = a2+b2−2abcos⁡γ\displaystyle a^2+b^2-2ab\cos\gamma