Equations and functions

For a linear function, the usual form is:

$y=ax+b$

A quadratic equation is written as:

$ax^2+bx+c=0$

If the two roots of the quadratic equation are $x_1$ and $x_2$, Vieta’s formulas give:

$x_1+x_2=-\cfrac{b}{a}$

$x_1x_2=\cfrac{c}{a}$

Trigonometric calculations

For an angle $\alpha$ in a right-triangle or coordinate setting, let:

$r=\sqrt{x^2+y^2}$

Then the basic trigonometric ratios are:

$sin(\alpha)=\cfrac{y}{r}$ sine

$cos(\alpha)=\cfrac{x}{r}$ cosine

$tan(\alpha)=\cfrac{y}{x}$ tangent

Special angles

<table> <thead> <tr> <th>Radians</th> <th>0</th> <th>$\cfrac{\pi}{2}$</th> <th>$\pi$</th> <th>$\cfrac{3\pi}{2}$</th> <th>$2\pi$</th> </tr> </thead> <tbody> <tr> <td>Degrees</td> <td>0°</td> <td>90°</td> <td>180°</td> <td>270°</td> <td>360°</td> </tr> <tr> <td>$sin(\alpha)$</td> <td>0</td> <td>1</td> <td>0</td> <td>-1</td> <td>0</td> </tr> <tr> <td>$cos(\alpha)$</td> <td>1</td> <td>0</td> <td>-1</td> <td>0</td> <td>1</td> </tr> <tr> <td>$tan(\alpha)$</td> <td>0</td> <td>undefined</td> <td>0</td> <td>undefined</td> <td>0</td> </tr> </tbody> </table>

Basic identities

$sin^2(\alpha)+cos^2(\alpha)=1$

$tan(\alpha)=\cfrac{sin(\alpha)}{cos(\alpha)}$

Reduction formulas

$sin(2k\pi+\alpha)=sin(\alpha)$

$cos(2k\pi+\alpha)=cos(\alpha)$

$tan(2k\pi+\alpha)=tan(\alpha)$

This is equivalent to rotating through $k$ full turns; the final angle still has the same relative position as $\alpha$.

$sin(\pi+\alpha)=-sin(\alpha)$

$cos(\pi+\alpha)=-cos(\alpha)$

$tan(\pi+\alpha)=tan(\alpha)$

$sin(\pi-\alpha)=sin(\alpha)$

$cos(\pi-\alpha)=-cos(\alpha)$

$tan(\pi-\alpha)=-tan(\alpha)$

Sum and difference formulas

For the sum of two angles:

$sin(\alpha+\beta)=sin(\alpha)cos(\beta)+cos(\alpha)sin(\beta)\qquad$ sine

$cos(\alpha+\beta)=cos(\alpha)cos(\beta)-sin(\alpha)sin(\beta)\qquad$ cosine

$tan(\alpha+\beta)=\cfrac{tan(\alpha)+tan(\beta)}{1-tan(\alpha)tan(\beta)}\qquad$ tangent

For a difference of two angles, reverse the signs inside the formulas: plus becomes minus, and minus becomes plus.

Double-angle formulas

$sin(2\alpha)=2sin(\alpha)cos(\alpha)$

$cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)=2cos^2(\alpha)-1=1-2sin^2(\alpha)$

$tan(2\alpha)=\cfrac{2tan(\alpha)}{1-tan^2(\alpha)}$

A useful way to remember the cosine double-angle forms is that whenever a sign appears on the right-hand side, it is a minus sign.

Sine-type functions

A sine-type function is commonly written as:

$y=Asin(\omega x+\phi)$

Here:

• $A$ is the amplitude.
• $\omega x+\phi$ is the phase.
• $\phi$ is the initial phase.
• $T=\cfrac{2\pi}{\omega}$ is the period.
• $f=\cfrac{1}{T}=\cfrac{\omega}{2\pi}$ is the frequency.

Solving triangles

Solving a triangle means finding its unknown sides and angles from the given conditions.

Area formula for a triangle

$S\triangle ABC=(\cfrac{1}{2}bc)sinA=(\cfrac{1}{2}ac)sinB=(\cfrac{1}{2}ab)sinC$

Sine rule

$\cfrac{a}{sinA}=\cfrac{b}{sinB}=\cfrac{c}{sinC}$

Cosine rule

$a^2=b^2+c^2-2bc\,cosA$

$b^2=a^2+c^2-2ac\,cosB$

$c^2=a^2+b^2-2ab\,cosC$

A simple way to remember the cosine rule is to look at the squared letter on the left. The two squared terms on the right are the other two side lengths. The cosine term uses the angle with the same capital letter as the side on the left.

In words: one side squared equals the sum of the squares of the other two sides, minus twice the product of those two sides times the cosine of the included angle.

The cosine rule can also be rearranged as:

$cosA=\cfrac{b^2+c^2-a^2}{2bc}$

$cosB=\cfrac{a^2+c^2-b^2}{2ac}$

$cosC=\cfrac{a^2+b^2-c^2}{2ab}$