Test of Mathematics for University Admission (TMUA) 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 Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Trigonometric Ratios and Rules

   3 topics

   • Right-Angled Trigonometry
     • Sine, cosine and tangent ratios
     • Exact values for special angles
   • Sine and Cosine Rules
     • Solving non-right-angled triangles
     • Ambiguous case of the sine rule
   • Area of a Triangle
     • Using half a b sin C
2. 2

   Trigonometric Functions and Graphs

   3 topics

   • Radian Measure
     • Converting degrees and radians
     • Arc length and sector area
   • Graphs of Trigonometric Functions
     • Sine, cosine and tangent curves
     • Period, amplitude and symmetry
   • Transformations of Trig Graphs
     • Translations and stretches applied to trig curves
3. 3

   Trigonometric Identities and Equations

   3 topics

   • Fundamental Identities
     • Pythagorean identity
     • tan as sin over cos
   • Solving Trigonometric Equations
     • Solutions within a given interval
     • Equations requiring identities
   • Compound and Double Angle Awareness
     • Recognising standard angle relationships

Trigonometry flashcards for Test of Mathematics for University Admission (TMUA)

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

1. In a right-angled triangle, how are $\sin\theta$, $\cos\theta$ and $\tan\theta$ defined in terms of the opposite, adjacent and hypotenuse?

   $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$ (remembered as SOH-CAH-TOA).

2. What is the relationship between $\tan\theta$, $\sin\theta$ and $\cos\theta$?

   $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, valid wherever $\cos\theta \neq 0$.

3. State the exact values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ for $\theta = 30^{\circ}$ (i.e. $\tfrac{\pi}{6}$).

   $\sin 30^{\circ} = \dfrac{1}{2}$, $\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}$, $\tan 30^{\circ} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$.

4. State the exact values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ for $\theta = 45^{\circ}$ (i.e. $\tfrac{\pi}{4}$).

   $\sin 45^{\circ} = \cos 45^{\circ} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$, and $\tan 45^{\circ} = 1$.

5. State the exact values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ for $\theta = 60^{\circ}$ (i.e. $\tfrac{\pi}{3}$).

   $\sin 60^{\circ} = \dfrac{\sqrt{3}}{2}$, $\cos 60^{\circ} = \dfrac{1}{2}$, $\tan 60^{\circ} = \sqrt{3}$.

6. What are the exact values of $\sin 0^{\circ}$, $\cos 0^{\circ}$, $\sin 90^{\circ}$ and $\cos 90^{\circ}$?

   $\sin 0^{\circ} = 0$, $\cos 0^{\circ} = 1$, $\sin 90^{\circ} = 1$, $\cos 90^{\circ} = 0$. (Also $\tan 0^{\circ} = 0$ and $\tan 90^{\circ}$ is undefined.)

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

   $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. The reciprocal form $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ is convenient when finding an angle.

8. When is it appropriate to use the Sine Rule rather than the Cosine Rule?

   Use the Sine Rule when you know two angles and any side (AAS/ASA), or two sides and a non-included angle (SSA). It pairs an angle with its opposite side.

9. State the Cosine Rule for finding side $a$ given sides $b, c$ and included angle $A$.

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

10. Rearrange the Cosine Rule to find an angle $A$ when all three sides $a, b, c$ are known.

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

11. When should the Cosine Rule be used instead of the Sine Rule?

    Use the Cosine Rule when you know two sides and the included angle (SAS) to find the third side, or when you know all three sides (SSS) to find an angle.

12. What is the 'ambiguous case' of the Sine Rule?

    In the SSA situation (two sides and a non-included angle), $\sin\theta = k$ can give two possible angles, $\theta$ and $180^{\circ} - \theta$, potentially yielding two valid triangles. Both solutions must be checked against the angle sum.

13. Give the formula for the area of a triangle using two sides and the included angle.

    $\text{Area} = \dfrac{1}{2}ab\sin C$, where $a$ and $b$ are two sides and $C$ is the angle between them.

14. How many radians correspond to a full turn, and what is the conversion between radians and degrees?

    A full turn is $2\pi$ radians $= 360^{\circ}$, so $\pi$ radians $= 180^{\circ}$. To convert degrees to radians multiply by $\dfrac{\pi}{180}$; to convert radians to degrees multiply by $\dfrac{180}{\pi}$.

15. Define one radian.

    One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius.

16. State the formula for the arc length $s$ of a sector of radius $r$ subtending angle $\theta$ in radians.

    $s = r\theta$ (with $\theta$ in radians).

17. State the formula for the area $A$ of a sector of radius $r$ subtending angle $\theta$ in radians.

    $A = \dfrac{1}{2}r^{2}\theta$ (with $\theta$ in radians).

18. Express $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, $90^{\circ}$ and $180^{\circ}$ in radians.

    $30^{\circ} = \dfrac{\pi}{6}$, $45^{\circ} = \dfrac{\pi}{4}$, $60^{\circ} = \dfrac{\pi}{3}$, $90^{\circ} = \dfrac{\pi}{2}$, $180^{\circ} = \pi$.

19. State the area of a segment of a circle of radius $r$ cut off by a chord subtending angle $\theta$ (radians) at the centre.

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

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

    Period $2\pi$ (or $360^{\circ}$), amplitude $1$, range $-1 \leq y \leq 1$. It passes through the origin and $\sin x = 0$ at integer multiples of $\pi$.

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