Trigonometry (Pure 1) — A-Level Mathematics Pure 1 Study Guide

For: A-Level Mathematics candidates sitting Paper 1 (Pure Mathematics 1).

Covers: Sine and cosine rules, area of a triangle formula, graphs of sine, cosine and tangent, core trigonometric identities, solving trigonometric equations, graph transformations, and exact values for special angles.

You should already know: IGCSE / Add-Maths algebra, sketching basic curves, solving linear and simple quadratic equations.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the A-Level Mathematics style for educational use. They are not reproductions of past Cambridge International examination papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official Cambridge mark schemes for grading conventions.

1. What Is Trigonometry (Pure 1)?

Trigonometry is the branch of mathematics that describes relationships between the side lengths and angles of triangles, extended to model periodic, cyclic phenomena such as waves, oscillations, and circular motion. For A-Level Mathematics Paper 1, this topic contributes 15-20% of total exam marks, and is often combined with coordinate geometry, calculus, or geometric proof questions to test cross-topic application.

2. Sine and cosine rules; area $\frac{1}{2}ab\sin C$

These rules apply to all non-right-angled and right-angled triangles, removing the limitation of basic SOHCAHTOA which only works for right-angled triangles. For a standard labelled triangle where side $a$ is opposite angle $A$, side $b$ opposite angle $B$, and side $c$ opposite angle $C$:

• Sine Rule: Relates side lengths to the sine of their opposite angles: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ Use this when you have 2 angles and 1 side, or 2 sides and a non-included angle. Note the ambiguous case: if you have 2 sides and a non-included acute angle, there may be 2 valid solutions for the missing angle (one acute, one obtuse, as $\sin x=\sin (180^{\circ }-x)$).
• Cosine Rule: Relates three sides to one included angle: $$a^2=b^2+c^2-2bc\cos A\text{or rearranged}\cos A=\frac{b^2+c^2-a^2}{2bc}$$ Use this when you have 3 sides, or 2 sides and the included angle between them.
• Area of a triangle: Uses two sides and the included angle between them: $$\text{Area}=\frac{1}{2}ab\sin C$$

Worked Example: A triangle has sides 6cm and 9cm, with an included angle of $45^{\circ }$. Calculate its area to 2 decimal places. Substitute into the area formula: $$\text{Area}=\frac{1}{2}\times 6\times 9\times \sin 45^{\circ }=27\times \frac{\sqrt{2}}{2}\approx 19.09{\text{ cm}}^2$$

Exam tip: Always confirm if your calculator is set to degrees or radians before solving triangle questions, as this is the most common avoidable error on this subtopic.

3. Graphs of $\sin ,\cos ,\tan$ — period, amplitude, asymptotes

Trigonometric functions are periodic, meaning they repeat their shape at regular intervals. Key definitions:

• Amplitude: The maximum distance from the midline of the graph to its peak or trough.
• Period: The smallest horizontal distance after which the graph repeats its full shape.
• Asymptote: A vertical line the graph approaches but never touches, where the function is undefined.

Worked Example: State the number of vertical asymptotes of $y=\tan (3x)$ for $0^{\circ }\le x\le 360^{\circ }$. Asymptotes occur when $3x=90^{\circ },270^{\circ },450^{\circ },630^{\circ },810^{\circ },990^{\circ }$, so $x=30^{\circ },90^{\circ },150^{\circ },210^{\circ },270^{\circ },330^{\circ }$: 6 asymptotes total.

4. Identities: ${\sin }^2\theta +{\cos }^2\theta =1$ and $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Identities are equations that hold true for all valid values of the input variable, and are used to simplify expressions or solve equations.

• The Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$ comes from applying Pythagoras' theorem to a right-angled triangle with hypotenuse 1 on the unit circle.
• The tangent identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$ holds for all $\theta$ where $\cos \theta \ne 0$ (i.e. no values where $\tan \theta$ is undefined).

Worked Example: Given $\cos \theta =-\frac{5}{13}$ and $\theta$ is obtuse, find the exact value of $\tan \theta$. First use the Pythagorean identity: $${\sin }^2\theta =1-{\cos }^2\theta =1-\frac{25}{169}=\frac{144}{169}$$ Since $\theta$ is obtuse, $\sin \theta$ is positive, so $\sin \theta =\frac{12}{13}$. Then: $$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\frac{12}{13}}{-\frac{5}{13}}=-\frac{12}{5}$$

Exam tip: These identities are not provided in the formula booklet, so you must memorize them for the exam.

5. Solving trig equations — principal value + general solution

Trigonometric equations have infinitely many solutions due to their periodicity. Questions will either ask for solutions in a specified range, or the general solution that describes all valid inputs.

• Principal value: The smallest positive or negative angle that satisfies the equation, usually in the range $0^{\circ }\le \alpha <360^{\circ }$ or $-\pi \le \alpha <\pi$.
• General solution formulas (for integer $n$):

1. If $\sin \theta =k$ where $|k|\le 1$: $\theta =360^{\circ }n+\alpha$ or $\theta =360^{\circ }n+180^{\circ }-\alpha$
2. If $\cos \theta =k$ where $|k|\le 1$: $\theta =360^{\circ }n\pm \alpha$
3. If $\tan \theta =k$ for any real $k$: $\theta =180^{\circ }n+\alpha$

Worked Example: Solve $2\cos (2x)=1$ for $0\le x\le \pi$ radians, then state the general solution. First rearrange: $\cos (2x)=0.5$. The principal value of $2x$ is $\frac{\pi }{3}$. The second solution for $2x$ in $0\le 2x\le 2\pi$ is $2\pi -\frac{\pi }{3}=\frac{5\pi }{3}$. So $x=\frac{\pi }{6},\frac{5\pi }{6}$. General solution: $2x=2\pi n\pm \frac{\pi }{3}$, so $x=\pi n\pm \frac{\pi }{6}$ for integer $n$.

6. Transformations of trig graphs — vertical / horizontal stretch and shift

Trig graphs follow the same transformation rules as all other functions. For a base function $y=f(x)$, the transformed function $y=af(k(x+d))+c$ has the following properties:

1. $a$: Vertical stretch by factor $a$ parallel to the y-axis, amplitude multiplied by $a$ (if $a$ is negative, the graph is reflected over the x-axis)
2. $k$: Horizontal stretch by factor $\frac{1}{k}$ parallel to the x-axis, period divided by $k$
3. $d$: Horizontal shift $d$ units to the left (if $d$ is positive) or $d$ units to the right (if $d$ is negative)
4. $c$: Vertical shift $c$ units up (if $c$ is positive) or $c$ units down (if $c$ is negative)

Worked Example: Describe the transformation sequence mapping $y=\sin x$ to $y=3\sin (2x-60^{\circ })+2$. First factor out the coefficient of $x$ inside the function: $y=3\sin \left(2\left(x-30^{\circ }\right)\right)+2$. The sequence is: 1) Vertical stretch by factor 3 parallel to the y-axis, 2) Horizontal stretch by factor $\frac{1}{2}$ parallel to the x-axis, 3) Shift 30° to the right, 4) Shift 2 units up.

Exam tip: Always factor out the coefficient of $x$ before describing horizontal shifts, otherwise you will miscalculate the shift size and lose marks.

7. Exact values for special angles (0°, 30°, 45°, 60°, 90° and radian equivalents)

Exam questions frequently ask for exact values of trig functions for special angles, which you must memorize to avoid losing marks. You can derive these values from two standard right-angled triangles:

1. 45-45-90 triangle: sides 1, 1, $\sqrt{2}$
2. 30-60-90 triangle: sides 1, $\sqrt{3}$, 2

Worked Example: Calculate the exact value of $\sin \frac{\pi }{4}\cos \frac{\pi }{6}-\cos \frac{\pi }{4}\sin \frac{\pi }{6}$. Substitute exact values: $$\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)-\left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}=\frac{\sqrt{6}-\sqrt{2}}{4}$$

8. Common Pitfalls (and how to avoid them)

• Wrong move: Using the sine rule for 2 sides and an included angle, or missing the ambiguous case when using the sine rule for non-included angles. Why it happens: Confusing use cases for sine and cosine rules. Correct move: Use the cosine rule for included angles; if you get an acute angle from the sine rule for a non-included angle, always check if the obtuse alternative ($180^{\circ }-\alpha$) gives a valid total angle sum below $180^{\circ }$.
• Wrong move: Trying to solve equations where $\sin \theta$ or $\cos \theta$ is outside the range $[-1,1]$, or not catching arithmetic errors when this happens. Why it happens: Forgetting the range of sine and cosine functions. Correct move: Always check if the value of $\sin \theta$ or $\cos \theta$ you calculate is between -1 and 1; if not, redo your arithmetic immediately.
• Wrong move: Using a period of $360^{\circ }$ for tan equations, missing half the solutions. Why it happens: Confusing the period of tan with sine and cosine. Correct move: For tan equations, add $180^{\circ }n$ to the principal value to get all solutions, not $360^{\circ }n$.
• Wrong move: Writing exact values with unrationalized denominators (e.g. $\frac{1}{\sqrt{2}}$ instead of $\frac{\sqrt{2}}{2}$). Why it happens: Rushing to finish answers. Correct move: Always rationalize denominators for exact value answers unless the question explicitly says otherwise.

9. Practice Questions (A-Level Mathematics Paper 1 Style)

Question 1

Triangle $PQR$ has $PQ=10$ cm, $QR=7$ cm, and $\angle QPR=40^{\circ }$. (a) Find the two possible values of $\angle PRQ$, correct to 1 decimal place. (b) Calculate the area of the larger of the two possible triangles, correct to 2 decimal places.

Solution 1

(a) Apply the sine rule: $\frac{PQ}{\sin R}=\frac{QR}{\sin P}$ $$\frac{10}{\sin R}=\frac{7}{\sin 40^{\circ }}⟹\sin R=\frac{10\times \sin 40^{\circ }}{7}\approx 0.918$$ Acute $R\approx 66.6^{\circ }$, obtuse $R=180^{\circ }-66.6^{\circ }=113.4^{\circ }$. Both are valid as $40^{\circ }+113.4^{\circ }=153.4^{\circ }<180^{\circ }$. (b) The larger triangle has the smaller angle $R=66.6^{\circ }$, so $\angle PQR=180^{\circ }-40^{\circ }-66.6^{\circ }=73.4^{\circ }$ $$\text{Area}=\frac{1}{2}\times PQ\times QR\times \sin Q=\frac{1}{2}\times 10\times 7\times \sin 73.4^{\circ }\approx 33.52{\text{ cm}}^2$$

Question 2

(a) Solve $3{\sin }^{2}x-\sin x=0$ for $0^{\circ }\le x\le 360^{\circ }$. (b) State the general solution for the equation.

Solution 2

(a) Factor the equation: $\sin x(3\sin x-1)=0$ Case 1: $\sin x=0$, solutions $x=0^{\circ },180^{\circ },360^{\circ }$ Case 2: $\sin x=\frac{1}{3}$, principal value $\approx 19.5^{\circ }$, second solution $180^{\circ }-19.5^{\circ }=160.5^{\circ }$ All solutions: $0^{\circ },19.5^{\circ },160.5^{\circ },180^{\circ },360^{\circ }$ (b) General solution: $x=180^{\circ }n$ or $x=360^{\circ }n+19.5^{\circ }$ or $x=360^{\circ }n+160.5^{\circ }$ for integer $n$.

Question 3

The function $g(x)=2\cos \left(3x+\frac{\pi }{2}\right)-1$ is a transformation of $f(x)=\cos x$. (a) Describe the sequence of transformations mapping $f(x)$ to $g(x)$. (b) State the range and period of $g(x)$.

Solution 3

(a) Rewrite $g(x)$ as $2\cos \left(3\left(x+\frac{\pi }{6}\right)\right)-1$. Transformation sequence:

1. Vertical stretch by factor 2 parallel to the y-axis
2. Horizontal stretch by factor $\frac{1}{3}$ parallel to the x-axis
3. Shift $\frac{\pi }{6}$ radians to the left
4. Shift 1 unit down (b) Range: Original range $[-1,1]$ → after vertical stretch $[-2,2]$ → after shift down 1: $[-3,1]$. Period: Original period $2\pi$ → after horizontal stretch by $\frac{1}{3}$: $\frac{2\pi }{3}$ radians.

10. Quick Reference Cheatsheet

11. What's Next

Mastering the trigonometry concepts in this guide is critical for success in later A-Level Mathematics topics. In Pure Mathematics 3, you will build on these foundations to learn reciprocal trigonometric functions, compound and double angle identities, and trigonometric differentiation and integration. Trigonometry also appears extensively in Mechanics papers for force resolution, projectile motion, and circular motion problems, so a strong grasp of these core rules will reduce your revision workload for subsequent papers significantly.

If you are stuck on any of the concepts, worked examples, or practice questions in this guide, you can ask Ollie for personalized explanations, additional practice problems, or step-by-step walkthroughs at any time by visiting Ollie. We also recommend practicing with official A-Level Mathematics Paper 1 past papers to familiarize yourself with exam timing, question phrasing, and marking conventions before your test.

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