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20/10/2021
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Some useful formulas
Cosine function

Some properties of the cosine function (cos) :

    \(\displaystyle cos(-x)=cos(x)\)

    \(\displaystyle cos(\pi+x)=-cos(x)\)

    \(\displaystyle cos(\pi-x)=-cos(x)\)

    \(\displaystyle cos(\frac{\pi}{2}+x)=-sin(x)\)

    \(\displaystyle cos(\frac{\pi}{2}-x)=sin(x)\)

    \(\displaystyle cos(x+y)=cos(x)*cos(y)−sin(x)*sin(y)\)

    \(\displaystyle cos(x-y)=cos(x)*cos(y)+sin(x)*sin(y)\)

    \(\displaystyle \begin{aligned}cos(2x)&=cos^2(x)-sin^2(x) \\
    &=1-2*sin^2(x) \\
    &=2*cos^2(x)-1\end{aligned}\)

    \(\displaystyle \begin{aligned}cos(3x)&=cos(x)*\left(1-4*sin^2(x)\right) \\
    &=cos(x)*\left(4*cos^2(x)-3\right)\end{aligned}\)

    \(\displaystyle cos^2(\frac{x}{2})=\frac{1+cos(x)}{2}\)

    \(\displaystyle cos(x)+cos(y)=2*cos\left(\frac{x+y}{2}\right)*cos\left(\frac{x-y}{2}\right)\)

    \(\displaystyle cos(x)-cos(y)=-2*sin\left(\frac{x+y}{2}\right)*sin\left(\frac{x-y}{2}\right)\)

    \(\displaystyle cos(x)*cos(y)=\frac{1}{2}\left(cos(x+y)+cos(x-y)\right)\)

    \(\displaystyle cos(x)*sin(y)=\frac{1}{2}\left(sin(x+y)-sin(x-y)\right)\)


Sine function

Some properties of the sine function (sin) :

    \(\displaystyle sin(-x)=-sin(x)\)

    \(\displaystyle sin(\pi+x)=-sin(x)\)

    \(\displaystyle sin(\pi-x)=sin(x)\)

    \(\displaystyle sin(\frac{\pi}{2}+x)=cos(x)\)

    \(\displaystyle sin(\frac{\pi}{2}-x)=cos(x)\)

    \(\displaystyle sin(x+y)=sin(x)*cos(y)+cos(x)*sin(y)\)

    \(\displaystyle sin(x-y)=sin(x)*cos(y)-cos(x)*sin(y)\)

    \(\displaystyle sin(2x)=2*sin(x)*cos(x)\)

    \(\displaystyle \begin{aligned}sin(3x)&=sin(x)\left(4*cos^2(x)-1\right) \\
    &=sin(x)*\left(3-4*sin^2(x)\right)\end{aligned}\)

    \(\displaystyle sin^2(\frac{x}{2})=\frac{1-cos(x)}{2}\)

    \(\displaystyle sin(x)+sin(y)=2*sin\left(\frac{x+y}{2}\right)*cos\left(\frac{x-y}{2}\right)\)

    \(\displaystyle sin(x)-sin(y)=2*cos\left(\frac{x+y}{2}\right)*sin\left(\frac{x-y}{2}\right)\)

    \(\displaystyle sin(x)*sin(y)=\frac{1}{2}\left(-cos(x+y)+cos(x-y)\right)\)

    \(\displaystyle sin(x)*cos(y)=\frac{1}{2}\left(sin(x+y)+sin(x-y)\right)\)


Tangent function

Some properties of the tangent function (tan) :

    \(\displaystyle tan(-x)=-tan(x)\)

    \(\displaystyle tan(\pi+x)=tan(x)\)

    \(\displaystyle tan(\pi-x)=-tan(x)\)

    \(\displaystyle tan(\frac{\pi}{2}+x)=-cotan(x)\)

    \(\displaystyle tan(\frac{\pi}{2}-x)=cotan(x)\)

    \(\displaystyle tan(x+y)=\frac{tan(x)+tan(y)}{1-tan(x)*tan(y)}\)

    \(\displaystyle tan(x-y)=\frac{tan(x)-tan(y)}{1+tan(x)*tan(y)}\)

    \(\displaystyle tan(2x)=\frac{2*tan(x)}{1-tan^2(x)}\)

    \(\displaystyle tan(3x)=\frac{tan(x)\left(3-tan^2(x)\right)}{1-3*tan^2(x)}\)

    \(\displaystyle tan^2(\frac{x}{2})=\frac{1-cos(x)}{1+cos(x)}\)

    \(\displaystyle \begin{aligned}tan(\frac{x}{2})&=\frac{1-cos(x)}{sin(x)} \\
    &=\frac{sin(x)}{1+cos(x)}\end{aligned}\)

    \(\displaystyle tan(x)+tan(y)=\frac{sin(x+y)}{cos(x)*cos(y)}\)

    \(\displaystyle tan(x)-tan(y)=\frac{sin(x-y)}{cos(x)*cos(y)}\)


If you have any comments or questions about the trigonometry formulas, you can discuss them in the forum: Discussion forums.

CHALLENGE : If you want to practice demonstrating any of these formulas, you can propose a proof in the forum: Discussion forums.

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