Mathematics Admissions Test (MAT) Trigonometry Flashcards

24 sample cards from the Trigonometry deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

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

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 $a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)$.

3. 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 $\tan\!\left(\dfrac{\pi}{2}-\theta\right)=\cot\theta$.

4. 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 $n$.

5. 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).

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

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

7. 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 $x$.

8. 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.

9. 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 $\arctan x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$.

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

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

11. 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 $\dfrac{2\pi}{n}$.

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

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

13. 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.

14. 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 $\tan$).

15. 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.

16. 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 $\cos(\omega t)=-1$.

17. 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 $t$.

18. 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.)

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

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

20. 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 $\cos\theta\approx 1-\dfrac{\theta^{2}}{2}$.

21. 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 $\sin x$.

22. 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: $-\sqrt{a^{2}+b^{2}}\le a\sin\theta+b\cos\theta\le\sqrt{a^{2}+b^{2}}$.

23. 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).

24. 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 $\dfrac{\sin x}{x}<1$.

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Trigonometry flashcards FAQ