ACT Math 8: Trigonometry

SOH-CAH-TOA

A mnemonic for right-triangle ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.

Reciprocal trigonometric functions

$\csc A=\frac{1}{\sin A}$, $\sec A=\frac{1}{\cos A}$, and $\cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}$.

Area of a triangle from two sides and their included angle

$\Delta=\frac{1}{2}ab\sin C$, where $C$ is the angle between sides $a$ and $b$.

Pythagorean trigonometric identity

$\sin^2\theta+\cos^2\theta=1$. It follows from the Pythagorean theorem applied to the unit circle.

Trigonometry

The branch of mathematics that studies relationships among angles and side lengths, especially through ratios in triangles.

How are the hypotenuse, adjacent side, and opposite side identified relative to an angle?

The hypotenuse is opposite the right angle. The adjacent side is the non-hypotenuse side touching the chosen angle, and the opposite side does not touch that angle.

Why do trigonometric ratios depend only on an angle in a right triangle?

Right triangles sharing an acute angle are similar, so corresponding side-length ratios are constant regardless of the triangle's size.

Sine of an acute angle in a right triangle

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine of an acute angle in a right triangle

$\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent of an acute angle in a right triangle

$\tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sin A}{\cos A}$.

What information is sufficient to determine an arbitrary triangle using trigonometric laws?

A triangle can generally be solved when two sides and their included angle, two angles and one side, or all three sides are known.

Radians and degrees

A full revolution is $360^\circ=2\pi$ radians. Convert using $\theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180}$ or $\theta_{\text{deg}}=\theta_{\text{rad}}\frac{180}{\pi}$.

Unit circle definition of sine and cosine

For an angle in standard position, if its terminal side intersects the unit circle at $(x,y)$, then $\cos\theta=x$ and $\sin\theta=y$.

Why does the unit circle extend trigonometric functions beyond right triangles?

The coordinates of the point where an angle's terminal side meets the unit circle define sine and cosine for positive and negative angles in all quadrants.

Exact trigonometric values at $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$

For sine, the values are $0,\frac{1}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},1$; for cosine, they are $1,\frac{\sqrt{3}}{2},\frac{\sqrt{2}}{2},\frac{1}{2},0$.

How can the signs of sine and cosine be determined from the unit circle?

Cosine is the $x$-coordinate and sine is the $y$-coordinate. Thus their signs follow the signs of the point's coordinates in each quadrant.

What happens to tangent at angles where cosine is zero?

Because $\tan\theta=\frac{\sin\theta}{\cos\theta}$, tangent is undefined when $\cos\theta=0$, such as at $\theta=\frac{\pi}{2}+n\pi$.

Period of sine and cosine

Both functions repeat every $2\pi$: $\sin(\theta+2\pi)=\sin\theta$ and $\cos(\theta+2\pi)=\cos\theta$.

Period of tangent and cotangent

Both tangent and cotangent repeat every $\pi$: $\tan(\theta+\pi)=\tan\theta$ and $\cot(\theta+\pi)=\cot\theta$, wherever defined.

Domains and ranges of sine and cosine

For real inputs, both have domain $(-\infty,\infty)$ and range $[-1,1]$.

Domains and ranges of tangent, secant, cosecant, and cotangent

Tangent excludes $\frac{\pi}{2}+n\pi$ and has range all real numbers. Cotangent excludes $n\pi$ and has range all real numbers. Secant and cosecant have range $(-\infty,-1]\cup[1,\infty)$, with secant excluding $\frac{\pi}{2}+n\pi$ and cosecant excluding $n\pi$.

Why do ordinary trigonometric functions not have global inverse functions?

They are periodic and therefore not one-to-one: different angles produce the same value. Restricting their domains creates invertible branches called principal-value inverse functions.

Principal-value range of $\arcsin x$

$\arcsin x$ is defined for $-1\le x\le1$ and returns an angle in $[-\frac{\pi}{2},\frac{\pi}{2}]$.

Principal-value range of $\arccos x$

$\arccos x$ is defined for $-1\le x\le1$ and returns an angle in $[0,\pi]$.

Principal-value range of $\arctan x$

$\arctan x$ accepts every real input and returns an angle in $(-\frac{\pi}{2},\frac{\pi}{2})$.

Why are radians the natural angle unit in calculus and series?

The standard derivatives and Maclaurin series, such as $\frac{d}{dx}\sin x=\cos x$, hold directly when $x$ is measured in radians.

Maclaurin series for sine

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$.

Maclaurin series for cosine

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$.

Chord function

The historical chord function is $\operatorname{crd}\theta=2\sin\left(\frac{\theta}{2}\right)$, equal to the length of a chord subtending angle $\theta$ in a unit circle.

Versine and haversine

The versine is $\operatorname{vers}\theta=1-\cos\theta=2\sin^2\left(\frac{\theta}{2}\right)$. The haversine is half the versine: $\operatorname{hav}\theta=\sin^2\left(\frac{\theta}{2}\right)$.

How can trigonometric functions model periodic physical phenomena?

Sine and cosine functions represent repeating quantities such as sound, light, and oscillations. Their amplitude describes the size of variation, while their period or frequency describes how rapidly the pattern repeats.

Fourier series

A Fourier series represents a periodic function as a sum of sine and cosine functions with different amplitudes and harmonically related frequencies.

Fourier transform

A Fourier transform expresses a function in terms of its frequency components, showing which frequencies contribute to a signal. This is useful for analyzing waves, communications, and spectra.

How is trigonometry used in navigation and surveying?

Measured angles can be combined with triangle relationships to determine distances, locations, areas, and relative directions. Modern systems such as GPS use triangulation or related geometric calculations.

How are trigonometric functions relevant to chemistry and physical science?

They describe periodic waves and oscillations, including light and sound, and appear in optics, spectroscopy-related wave analysis, crystallography, imaging, and models of molecular or electronic behavior.

Law of sines

For sides $a,b,c$ opposite angles $A,B,C$, respectively, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$, where $R$ is the circumradius.

When is the law of sines especially useful?

It is useful when a triangle includes two known angles and a side or two known sides with a known non-included opposite angle, allowing corresponding sides and angles to be related.

Law of cosines

For a triangle with sides $a,b,c$ and angle $C$ between $a$ and $b$, $c^2=a^2+b^2-2ab\cos C$. It generalizes the Pythagorean theorem to non-right triangles.

How can the law of cosines be used to find an unknown angle?

Rearrange it, for example, as $\cos C=\frac{a^2+b^2-c^2}{2ab}$, then use $C=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)$.

Derived Pythagorean trigonometric identities

Dividing $\sin^2\theta+\cos^2\theta=1$ by $\cos^2\theta$ gives $\tan^2\theta+1=\sec^2\theta$; dividing by $\sin^2\theta$ gives $\cot^2\theta+1=\csc^2\theta$.

Euler's formula for trigonometric functions

$e^{i\theta}=\cos\theta+i\sin\theta$, where $i^2=-1$. It connects complex exponentials with rotations and trigonometric functions.

Sine and cosine in terms of complex exponentials

$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$ and $\cos x=\frac{e^{ix}+e^{-ix}}{2}$.