Mathematics Admissions Test (MAT) Trigonometry Syllabus

Trigonometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Trigonometry in Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Trigonometric Functions and Graphs

   3 topics

   • Definitions and the Unit Circle
     • Sine, cosine and tangent for any angle
     • Degrees and radians
     • Exact values for standard angles
   • Graphs of Trigonometric Functions
     • Shape, period and amplitude of sine, cosine and tangent
     • Transformations applied to trigonometric graphs
   • Triangle Rules
     • Sine rule and cosine rule
     • Area of a triangle using one half ab sin C
2. 2

   Identities and Equations

   3 topics

   • Fundamental Identities
     • The Pythagorean identity
     • The tan identity as sine over cosine
   • Solving Trigonometric Equations
     • Finding all solutions in a given interval
     • Equations reducible to quadratics in a trig function
     • Using identities to simplify before solving
   • Trigonometry in Geometry
     • Applying trig to lengths, angles and areas in figures
3. 3

   Trigonometry in Problem Solving

   3 topics

   • Modelling with Trigonometric Functions
     • Periodic behaviour and amplitude in applied contexts
     • Interpreting maxima and minima of a trig model
   • Combining Trigonometry with Other Methods
     • Using trig within coordinate geometry
     • Trigonometric substitution in algebraic problems
   • Estimation and Bounding
     • Bounding trigonometric expressions to reason about ranges

Trigonometry flashcards for Mathematics Admissions Test (MAT)

20 of 50 cards from the Trigonometry deck — real questions with worked answers.

1. On the unit circle, how are $\cos\theta$ and $\sin\theta$ defined for a point $P$ at angle $\theta$ measured anticlockwise from the positive $x$-axis?

   $P$ has coordinates $(\cos\theta,\sin\theta)$, so $\cos\theta$ is the $x$-coordinate and $\sin\theta$ is the $y$-coordinate of the point on the unit circle.

2. How is $\tan\theta$ defined in terms of $\sin\theta$ and $\cos\theta$, and where is it undefined?

   $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. It is undefined wherever $\cos\theta=0$, i.e. at $\theta=\dfrac{\pi}{2}+k\pi$ for integer $k$.

3. How do you convert an angle from degrees to radians, and from radians to degrees?

   Degrees to radians: multiply by $\dfrac{\pi}{180}$. Radians to degrees: multiply by $\dfrac{180}{\pi}$. So $180^{\circ}=\pi$ radians.

4. State the exact values of $\sin$, $\cos$ and $\tan$ at $\theta=\dfrac{\pi}{6}$ ($30^{\circ}$).

   $\sin\dfrac{\pi}{6}=\dfrac{1}{2}$, $\cos\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{2}$, $\tan\dfrac{\pi}{6}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}$.

5. State the exact values of $\sin$, $\cos$ and $\tan$ at $\theta=\dfrac{\pi}{4}$ ($45^{\circ}$).

   $\sin\dfrac{\pi}{4}=\cos\dfrac{\pi}{4}=\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}$, and $\tan\dfrac{\pi}{4}=1$.

6. State the exact values of $\sin$, $\cos$ and $\tan$ at $\theta=\dfrac{\pi}{3}$ ($60^{\circ}$).

   $\sin\dfrac{\pi}{3}=\dfrac{\sqrt{3}}{2}$, $\cos\dfrac{\pi}{3}=\dfrac{1}{2}$, $\tan\dfrac{\pi}{3}=\sqrt{3}$.

7. What are the values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ at $\theta=0$ and at $\theta=\dfrac{\pi}{2}$?

   At $\theta=0$: $\sin 0=0$, $\cos 0=1$, $\tan 0=0$. At $\theta=\dfrac{\pi}{2}$: $\sin\dfrac{\pi}{2}=1$, $\cos\dfrac{\pi}{2}=0$, $\tan\dfrac{\pi}{2}$ is undefined.

8. Which trigonometric functions are positive in each quadrant (the CAST/ASTC rule)?

   Quadrant I: All positive. Quadrant II: only Sine. Quadrant III: only Tangent. Quadrant IV: only Cosine.

9. State the sign-change (negative angle) identities for $\sin$, $\cos$ and $\tan$, classifying each as odd or even.

   $\sin(-\theta)=-\sin\theta$ (odd), $\cos(-\theta)=\cos\theta$ (even), $\tan(-\theta)=-\tan\theta$ (odd).

10. For the arc and sector of a circle of radius $r$ subtending angle $\theta$ (in radians), give the arc length and sector area.

    Arc length $s=r\theta$; sector area $A=\dfrac{1}{2}r^{2}\theta$.

11. What are the period, amplitude and range of $y=\sin x$ and $y=\cos x$?

    Both have period $2\pi$, amplitude $1$, and range $[-1,1]$. ($\sin$ passes through the origin; $\cos$ peaks at $x=0$.)

12. What are the period, range and asymptotes of $y=\tan x$?

    Period $\pi$, range all real numbers $(-\infty,\infty)$, with vertical asymptotes at $x=\dfrac{\pi}{2}+k\pi$.

13. For $y=a\sin(bx+c)+d$, what do $a$, $b$, $c$ and $d$ each control?

    $|a|$ is the amplitude, the period is $\dfrac{2\pi}{|b|}$, $c$ produces a horizontal (phase) shift of $-\dfrac{c}{b}$, and $d$ is the vertical shift (midline $y=d$).

14. How are the graphs of $y=\sin x$ and $y=\cos x$ related by a translation?

    $\cos x=\sin\!\left(x+\dfrac{\pi}{2}\right)$, so the cosine graph is the sine graph translated left by $\dfrac{\pi}{2}$; equivalently $\sin x=\cos\!\left(x-\dfrac{\pi}{2}\right)$.

15. State the Sine Rule for a triangle with sides $a,b,c$ opposite angles $A,B,C$.

    $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$, where $R$ is the circumradius.

16. State the Cosine Rule and rearrange it to find an angle.

    $a^{2}=b^{2}+c^{2}-2bc\cos A$. Rearranged: $\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2bc}$.

17. Give two formulas for the area of a triangle using trigonometry.

    $\text{Area}=\dfrac{1}{2}ab\sin C$ (two sides and included angle). Heron's formula: $\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$ with $s=\dfrac{a+b+c}{2}$.

18. When solving a triangle with the Sine Rule, when does the 'ambiguous case' arise and how is it resolved?

    It arises when finding an angle from a given side-side-angle (SSA) configuration: $\sin\theta=k$ may give two valid angles $\theta$ and $180^{\circ}-\theta$. Check which satisfy the triangle's angle sum $180^{\circ}$ and side ordering.

19. State the Pythagorean identity relating $\sin$ and $\cos$, and its two derived forms.

    $\sin^{2}\theta+\cos^{2}\theta=1$. Dividing by $\cos^{2}\theta$: $1+\tan^{2}\theta=\sec^{2}\theta$. Dividing by $\sin^{2}\theta$: $1+\cot^{2}\theta=\csc^{2}\theta$.

20. Define the reciprocal trigonometric functions $\sec\theta$, $\csc\theta$ (cosec) and $\cot\theta$.

    $\sec\theta=\dfrac{1}{\cos\theta}$, $\csc\theta=\dfrac{1}{\sin\theta}$, $\cot\theta=\dfrac{1}{\tan\theta}=\dfrac{\cos\theta}{\sin\theta}$.

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