Mathematics Admissions Test (MAT) Trigonometry Syllabus

State the sine addition formulas $\sin(A\pm B)$.

$\sin(A+B)=\sin A\cos B+\cos A\sin B$ and $\sin(A-B)=\sin A\cos B-\cos A\sin B$.

State the cosine addition formulas $\cos(A\pm B)$.

$\cos(A+B)=\cos A\cos B-\sin A\sin B$ and $\cos(A-B)=\cos A\cos B+\sin A\sin B$.

State the tangent addition formula $\tan(A+B)$.

$\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}$.

State the double-angle formula for $\sin 2\theta$.

$\sin 2\theta=2\sin\theta\cos\theta$.

State the three equivalent forms of the double-angle formula for $\cos 2\theta$.

$\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta$.

State the double-angle formula for $\tan 2\theta$.

$\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^{2}\theta}$.

Give the half-angle (power-reduction) formulas for $\cos^{2}\theta$ and $\sin^{2}\theta$.

$\cos^{2}\theta=\dfrac{1+\cos 2\theta}{2}$ and $\sin^{2}\theta=\dfrac{1-\cos 2\theta}{2}$.

How do you write $a\sin\theta+b\cos\theta$ in the single form $R\sin(\theta+\alpha)$?

Take $R=\sqrt{a^{2}+b^{2}}$ and $\alpha=\arctan\!\left(\dfrac{b}{a}\right)$ (with $a=R\cos\alpha$, $b=R\sin\alpha$), giving $a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)$.

State the complementary (co-function) identities relating $\sin$ and $\cos$.

$\sin\!\left(\dfrac{\pi}{2}-\theta\right)=\cos\theta$ and $\cos\!\left(\dfrac{\pi}{2}-\theta\right)=\sin\theta$; likewise $\tan\!\left(\dfrac{\pi}{2}-\theta\right)=\cot\theta$.

To solve $\sin x=k$ for all solutions in $[0,2\pi)$, what is the general method?

Find the principal value $x_1=\arcsin k$. The second solution is $x_2=\pi-x_1$. All solutions are $x=x_1+2\pi n$ and $x=\pi-x_1+2\pi n$ for integer $n$.

To solve $\cos x=k$, how do you generate all solutions?

Principal value $x_1=\arccos k$ gives solutions $x=\pm x_1+2\pi n$ for integer $n$ (since cosine is even, $-x_1$ also works).

To solve $\tan x=k$, what is the general solution?

$x=\arctan k+\pi n$ for integer $n$, since the tangent function has period $\pi$.

When solving an equation like $\sin(2x)=k$ over $0\le x<2\pi$, why must you adjust the working interval?

If $0\le x<2\pi$ then the argument $2x$ ranges over $0\le 2x<4\pi$, so you must find all solutions for $2x$ across that wider interval before dividing by $2$ to recover $x$.

What is a standard strategy for solving a trig equation containing both $\sin^{2}x$ and $\cos x$ (e.g. $2\sin^{2}x+\cos x=1$)?

Use $\sin^{2}x=1-\cos^{2}x$ to write everything in terms of $\cos x$, obtaining a quadratic in $\cos x$, then solve and apply the inverse function to each root.

What ranges (principal value ranges) do $\arcsin x$, $\arccos x$ and $\arctan x$ take?

$\arcsin x\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$, $\arccos x\in[0,\pi]$, and $\arctan x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$.

In a right-angled triangle, what do the ratios SOH-CAH-TOA represent?

$\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$.

In a regular $n$-sided polygon inscribed in a circle of radius $r$, what is its area in terms of $n$ and $r$?

$\text{Area}=\dfrac{1}{2}n r^{2}\sin\!\left(\dfrac{2\pi}{n}\right)$, formed from $n$ isosceles triangles each with apex angle $\dfrac{2\pi}{n}$.

For a chord subtending angle $\theta$ at the centre of a circle of radius $r$, what is the chord length?

Chord length $=2r\sin\!\left(\dfrac{\theta}{2}\right)$.

What is the area of a circular segment cut off by a chord subtending central angle $\theta$ (radians) in a circle of radius $r$?

Segment area $=\dfrac{1}{2}r^{2}(\theta-\sin\theta)$, i.e. sector area minus triangle area.

In a 3D problem, how do you find the angle between a line and a plane?

Project the line onto the plane; the required angle is between the line and its projection. Compute it in the right-angled triangle formed by the line, its projection, and the perpendicular from the line's end to the plane (using $\sin$, $\cos$, or $\tan$).

In modelling, what do the amplitude and period of $y=A\sin(\omega t)+c$ represent physically (e.g. tides, oscillations)?

$A$ is the maximum deviation from the mean level $c$; the period $T=\dfrac{2\pi}{\omega}$ is the time for one complete cycle, and $\omega$ is the angular frequency.

For a sinusoidal model $h(t)=A\cos(\omega t)+c$ of a wheel/tide, how do you find the maximum and minimum values and when they occur?

Maximum $=c+|A|$ and minimum $=c-|A|$ (since $\cos$ ranges in $[-1,1]$). For $A>0$, the maximum occurs when $\cos(\omega t)=1$ (e.g. $t=0$) and minimum when $\cos(\omega t)=-1$.

How do you find the times when a sinusoidal model $A\sin(\omega t+\phi)+c$ reaches a given level $L$?

Set $A\sin(\omega t+\phi)+c=L$, solve $\sin(\omega t+\phi)=\dfrac{L-c}{A}$, find all solutions for the argument in the relevant range, then solve for $t$.

What is the derivative of $\sin x$, $\cos x$ and $\tan x$ (used in combining trig with calculus)?

$\dfrac{d}{dx}\sin x=\cos x$, $\dfrac{d}{dx}\cos x=-\sin x$, $\dfrac{d}{dx}\tan x=\sec^{2}x$. (Valid when $x$ is in radians.)

What are the integrals $\displaystyle\int\sin x\,dx$ and $\displaystyle\int\cos x\,dx$?

$\displaystyle\int\sin x\,dx=-\cos x+C$ and $\displaystyle\int\cos x\,dx=\sin x+C$.

What is the small-angle approximation for $\sin\theta$, $\cos\theta$ and $\tan\theta$ when $\theta$ (radians) is near $0$?

$\sin\theta\approx\theta$, $\tan\theta\approx\theta$, and $\cos\theta\approx 1-\dfrac{\theta^{2}}{2}$.

What is the value of $\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}$, and why is it useful?

$\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}=1$ (with $x$ in radians). It underlies the small-angle approximation $\sin x\approx x$ and the derivative of $\sin x$.

How can bounding the sine and cosine functions help when analysing an expression like $a\sin\theta+b\cos\theta$?

Since $a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)$ with $R=\sqrt{a^{2}+b^{2}}$, the expression is bounded: $-\sqrt{a^{2}+b^{2}}\le a\sin\theta+b\cos\theta\le\sqrt{a^{2}+b^{2}}$.

How do you find the maximum and minimum of a function such as $\dfrac{1}{3+\sin x}$ using bounds?

Since $-1\le\sin x\le 1$, the denominator satisfies $2\le 3+\sin x\le 4$. Hence the function ranges between $\dfrac{1}{4}$ (minimum) and $\dfrac{1}{2}$ (maximum).

Compare $\sin x$ and $x$ for $x>0$: which is larger, and how is this used to bound expressions?

For all $x>0$, $\sin x<x$ (and $0<x\le\tan x$ for $0<x<\tfrac{\pi}{2}$). These inequalities let you bound or estimate areas, sums, and limits, e.g. confirming $\dfrac{\sin x}{x}<1$.