CME 102 - Ordinary Differential Equations for Engineers

Trigonometry refresher
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By Afshine Amidi and Shervine Amidi

Definitions

Trigonometric functions The following common trigonometric functions are $2\pi$-periodic and are defined as follows:

Function Domain Image Definition Derivative
Cosine $\theta\in\mathbb{R}$ $\cos(\theta)\in[-1,1]$ $\cos(\theta)=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}$ $\displaystyle\frac{d\cos(\theta)}{d\theta}=-\sin(\theta)$
Sine $\theta\in\mathbb{R}$ $\sin(\theta)\in[-1,1]$ $\sin(\theta)=\frac{\textrm{opposite}}{\textrm{hypotenuse}}$ $\displaystyle\frac{d\sin(\theta)}{d\theta}=\cos(\theta)$
Tangent $\theta\in\mathbb{R}\backslash\{2k\pi\}$ $\tan(\theta)\in]-\infty,+\infty[$ $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{\textrm{opposite}}{\textrm{adjacent}}$ $\displaystyle\frac{d\tan(\theta)}{d\theta}=1+\tan^2(\theta)$

Euler's formula The following formula establishes a fundamental relationship between the trigonometric functions and the complex exponential function as follows:

\[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\]

Therefore, we have:

\[\boxed{\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}}\quad\textrm{and}\quad\boxed{\sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i}}\quad\textrm{and}\quad\boxed{\tan(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{i(e^{i\theta}+e^{-i\theta})}}\]

Inverse trigonometric functions The common inverse trigonometric functions are defined as follows:

Function Domain Image Definition Derivative
Arccosine $x\in[-1,1]$ $\textrm{arccos}(x)\in[0,\pi]$ $\cos(\textrm{arccos}(x))=x$ $\displaystyle\frac{d\textrm{arccos}(x)}{dx}=-\frac{1}{\sqrt{1-x^2}}$
Arcsine $x\in[-1,1]$ $\textrm{arcsin}(x)\in[-\frac{\pi}{2},\frac{\pi}{2}]$ $\sin(\textrm{arcsin}(x))=x$ $\displaystyle\frac{d\textrm{arcsin}(x)}{dx}=\frac{1}{\sqrt{1-x^2}}$
Arctangent $x\in]-\infty,+\infty[$ $\textrm{arctan}(x)\in]-\frac{\pi}{2},\frac{\pi}{2}[$ $\tan(\textrm{arctan}(x))=x$ $\displaystyle\frac{d\textrm{arctan}(x)}{dx}=\frac{1}{1+x^2}$

Trigonometric identities

Pythagorean identity The following identity is commonly used:

\[\forall \theta,\quad\boxed{\cos^2(\theta)+\sin^2(\theta)=1}\]

Inverse trigonometric identities The following identities are commonly used:

\[\forall x, \quad\boxed{\textrm{arccos}(x)+\textrm{arcsin}(x)=\frac{\pi}{2}}\quad\textrm{and}\quad\boxed{\textrm{arctan}(x)+\textrm{arctan}\left(\frac{1}{x}\right)=\left\{\begin{array}{cc}\displaystyle\frac{\pi}{2}&(x>0)\\&\\-\displaystyle\frac{\pi}{2}&(x<0)\end{array}\right.}\]

Addition formulas The following identities are commonly used:

Name Formula
Cosine addition $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$
Sine addition $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$
Tangent addition $\displaystyle\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$

Product-to-sum and sum-to-product identities The following identities are commonly used:

Name Formula
Product-to-sum $\displaystyle\cos(a)\cos(b)=\frac{1}{2}(\cos(a-b)+\cos(a+b))$
$\displaystyle\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))$
$\displaystyle\sin(a)\cos(b)=\frac{1}{2}(\sin(a+b)+\sin(a-b))$
$\displaystyle\cos(a)\sin(b)=\frac{1}{2}(\sin(a+b)-\sin(a-b))$
$\displaystyle\tan(a)\tan(b)=\frac{\cos(a-b)-\cos(a+b)}{\cos(a-b)+\cos(a+b)}$
Sum-to-product $\displaystyle\cos(a)+\cos(b)=2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$
$\displaystyle\cos(a)-\cos(b)=-2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$
$\displaystyle\sin(a)+\sin(b)=2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$
$\displaystyle\sin(a)-\sin(b)=2\sin\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)$

Symmetry identities The following identities are commonly used:

By $\alpha=0$ By $\alpha=\frac{\pi}{4}$ By $\alpha=\frac{\pi}{2}$
$\displaystyle\cos\left(-\theta\right)=\cos(\theta)$ $\displaystyle\cos\left(\frac{\pi}{2}-\theta\right)=\sin(\theta)$ $\displaystyle\cos\left(\pi-\theta\right)=-\cos(\theta)$
$\displaystyle\sin\left(-\theta\right)=-\sin(\theta)$ $\displaystyle\sin\left(\frac{\pi}{2}-\theta\right)=\cos(\theta)$ $\displaystyle\sin\left(\pi-\theta\right)=\sin(\theta)$
$\displaystyle\tan\left(-\theta\right)=-\tan(\theta)$ $\displaystyle\tan\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\tan(\theta)}$ $\displaystyle\tan\left(\pi-\theta\right)=-\tan(\theta)$

Shift identities The following identities are commonly used:

By $\frac{\pi}{2}$ By $\pi$
$\displaystyle\cos\left(\theta+\frac{\pi}{2}\right)=-\sin(\theta)$ $\displaystyle\cos\left(\theta+\pi\right)=-\cos(\theta)$
$\displaystyle\sin\left(\theta+\frac{\pi}{2}\right)=\cos(\theta)$ $\displaystyle\sin\left(\theta+\pi\right)=-\sin(\theta)$
$\displaystyle\tan\left(\theta+\frac{\pi}{2}\right)=-\frac{1}{\tan(\theta)}$ $\displaystyle\tan\left(\theta+\pi\right)=\tan(\theta)$

Miscellaneous

Kashani theorem The Kashani theorem, also known as the law of cosines, states that in a triangle, the lengths $a$, $b$, $c$ and the angle $\gamma$ between sides of length $a$ and $b$ satisfy the following equation:

\[\boxed{c^2=a^2+b^2-2ab\cos(\gamma)}\]

Remark: for $\gamma=\frac{\pi}{2}$, the triangle is right and the identity is the Pythagorean theorem.


Law of sines In a given triangle of lengths $a,b,c$ and opposite angles $A,B,C$, the law of sines states that we have:

\[\boxed{\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}}\]

Values for common angles The following table sums up the values for common angles to have in mind:

Angle $\theta$ (radians $\leftrightarrow$ degrees) $\cos(\theta)$ $\sin(\theta)$ $\tan(\theta)$
$\displaystyle0\leftrightarrow0^{\circ}$ $\displaystyle1$ $\displaystyle0$ $\displaystyle0$
$\displaystyle\frac{\pi}{6}\leftrightarrow30^{\circ}$ $\displaystyle\frac{\sqrt{3}}{2}$ $\displaystyle\frac{1}{2}$ $\displaystyle\frac{\sqrt{3}}{3}$
$\displaystyle\frac{\pi}{4}\leftrightarrow45^{\circ}$ $\displaystyle\frac{\sqrt{2}}{2}$ $\displaystyle\frac{\sqrt{2}}{2}$ $\displaystyle1$
$\displaystyle\frac{\pi}{3}\leftrightarrow60^{\circ}$ $\displaystyle\frac{1}{2}$ $\displaystyle\frac{\sqrt{3}}{2}$ $\displaystyle\sqrt{3}$
$\displaystyle\frac{\pi}{2}\leftrightarrow90^{\circ}$ $\displaystyle0$ $\displaystyle1$ $\infty$