O Level (Sec 3 & 4)Additional Mathematics
Trigonometric Functions
Master O-Level A Math trigonometric functions with clear notes on sine, cosine, tangent, identities, compound and double-angle formulae, R-formula and trigonometric equations. Includes a free worksheet and worked solutions for Singapore students.
By Timothy Gan29 November 2021Singapore O-Level syllabus
Free revision pack
Use the notes online, then download the worksheet to practise without distractions.
- 7 concept sections
- 5 worked questions
- 12 video lessons
7
Concept sections
5
Practice questions
5
Concept videos
7
Solution videos
Start here
Understanding Trigonometric Functions
Trigonometric functions connect angles to ratios. In O-Level Additional Mathematics, $\sin x$, $\cos x$ and $\tan x$ appear in identities, equations, graphs and later in calculus.
The main skill is not memorising formulas in isolation. Some identities are printed in the O-Level A-Math formula sheet, but you still need to recognise which identity changes the form of an expression, which angle range affects the answer, and when a trigonometric equation has more than one solution.
This guide gives you a compact revision path: start with the three core functions, then build towards identities, compound-angle formulae, double-angle formulae, R-formula and equation solving.
Video lesson
Study Guide - Trigonometric Ratios of Special Angles
1
Concept
The Three Basic Trigonometric Functions
For a right-angled triangle, the three main ratios are:
$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
A useful memory aid is SOH-CAH-TOA. In A Math, these ratios are extended beyond acute angles, so you must also understand signs in different quadrants and how angles repeat across cycles.
2
Concept
Basic Angles and the ASTC Rule
For angles outside the first quadrant, first find the basic angle between $0^\circ$ and $90^\circ$. Then use the quadrant to determine the sign of each trigonometric ratio.
The mnemonic ASTC gives the positive ratios as you move anticlockwise through the quadrants: All ratios in Quadrant I, Sine in Quadrant II, Tangent in Quadrant III and Cosine in Quadrant IV.
Always reduce angles outside one full revolution before applying the rule, and keep degree and radian notation consistent.
Video lesson
ASTC Rule, Part 1
Video lesson
ASTC Rule, Part 2
3
Concept
Reciprocal Trigonometric Ratios
The three reciprocal ratios are $\csc \theta=\frac{1}{\sin \theta}$, $\sec \theta=\frac{1}{\cos \theta}$ and $\cot \theta=\frac{1}{\tan \theta}$.
They can also be read directly from a right-angled triangle: cosecant is hypotenuse over opposite, secant is hypotenuse over adjacent, and cotangent is adjacent over opposite. These relationships are useful when a question gives one ratio and asks for the others.
Video lesson
Study Guide - Cotangent, Secant and Cosecant Ratios
4
Concept
Core Identities
The most important Pythagorean identity is:
$\sin^2 x + \cos^2 x = 1$
Dividing by $\cos^2 x$ gives:
$\tan^2 x + 1 = \sec^2 x$
Dividing by $\sin^2 x$ gives:
$1 + \cot^2 x = \csc^2 x$
At O-Level, the first two forms are especially useful. They help you simplify expressions, prove identities and convert a question into a single trigonometric function.
5
Concept
Compound and Double-Angle Formulae
Compound-angle formulae allow you to expand angles such as $A+B$ and $A-B$:
$\sin(A+B)=\sin A\cos B+\cos A\sin B$
$\cos(A+B)=\cos A\cos B-\sin A\sin B$
$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
Double-angle formulae come from setting $A=B$:
$\sin 2A=2\sin A\cos A$
$\cos 2A=\cos^2 A-\sin^2 A=2\cos^2 A-1=1-2\sin^2 A$
$\tan 2A=\frac{2\tan A}{1-\tan^2 A}$
6
Concept
R-Formula: Amplitude–Phase Form
The R-formula rewrites a linear combination of sine and cosine as one trigonometric function. Two useful forms are:
$a\sin x+b\cos x=R\sin(x+\alpha)$
$a\cos x+b\sin x=R\cos(x-\alpha)$
where $R=\sqrt{a^2+b^2}$. Always expand the form you chose and compare coefficients; this prevents sign errors.
For the sine form,
$R\sin(x+\alpha)=R\sin x\cos\alpha+R\cos x\sin\alpha,$
so $R\cos\alpha=a$ and $R\sin\alpha=b$. Hence $\tan\alpha=\frac{b}{a}$, with the quadrant chosen from the signs of the two coefficient equations.
Worked example
Express $3\sin x+4\cos x$ as $R\sin(x+\alpha)$.
$R=\sqrt{3^2+4^2}=5,$
$5\cos\alpha=3,\qquad5\sin\alpha=4,$
so $\alpha=\tan^{-1}(4/3)$ and
$3\sin x+4\cos x=5\sin\left(x+\tan^{-1}\frac{4}{3}\right).$
Because $-1\le\sin(x+\alpha)\le1$, the expression has maximum $5$ and minimum $-5$. If a constant is added, such as $7+3\sin x+4\cos x$, the range becomes $2\le y\le12$.
Exam checklist
- Match the requested sine or cosine form.
- Find $R$ from $\sqrt{a^2+b^2}$.
- Expand and compare both coefficients.
- Use the coefficient signs to place $\alpha$ in the correct quadrant.
- For an equation, solve the single trig function and list every solution in the stated interval.
The most common error is using $\tan\alpha=b/a$ without checking the signs. Coefficient comparison is the reliable method.
7
Concept
Solving Trigonometric Equations
A trigonometric equation can have several answers in a given interval. The safest process is:
- Rearrange into a standard form such as $\sin x=k$, $\cos x=k$ or $\tan x=k$.
- Find the basic angle.
- Use quadrant signs and the required interval to list all valid answers.
- Check whether the question uses degrees or radians.
For example, if $\sin x=\frac{1}{2}$ for $0^\circ \le x \le 360^\circ$, the answers are $30^\circ$ and $150^\circ$ because sine is positive in Quadrants I and II.
For inverse functions, use the principal-value ranges expected by your syllabus and calculator: $0\le x\le\pi$ for $\cos^{-1}$, and $-\frac{\pi}{2}\le x\le\frac{\pi}{2}$ for $\sin^{-1}$ and $\tan^{-1}$, with endpoints handled according to the function's domain.
Video lesson
Study Guide - Principal Values
Guided practice
Practice Questions with Video Solutions
Attempt each question before opening the solution. Then compare your method with the worked steps and video explanation.
Question 1Video solution
Exact Values of Special Angles
Question
Without using a calculator, find the exact value of $\frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{3}}$.
Video lesson
Study Guide - Trigonometric Functions Question 1
Step-by-step solution
- 1Use the exact values $\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}$ and $\cos \frac{\pi}{3}=\frac{1}{2}$.
- 2$\frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{3}}=\frac{1/\sqrt{2}}{1/2}=\frac{2}{\sqrt{2}}$.
- 3Rationalise and simplify: $\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\sqrt{2}$.
Final answer
$\sqrt{2}$
Question 2Video solution
Basic Angles and Quadrant Signs
Question
Express $\sin \theta$, $\cos \theta$ and $\tan \theta$ in terms of the ratios of the basic angle when $\theta=140^\circ$.
Video lesson
Study Guide - Trigonometric Functions Question 2
Step-by-step solution
- 1$140^\circ$ lies in Quadrant II, so the basic angle is $180^\circ-140^\circ=40^\circ$.
- 2In Quadrant II, sine is positive while cosine and tangent are negative.
- 3Therefore $\sin140^\circ=\sin40^\circ$, $\cos140^\circ=-\cos40^\circ$ and $\tan140^\circ=-\tan40^\circ$.
Final answer
$\sin140^\circ=\sin40^\circ$, $\cos140^\circ=-\cos40^\circ$, $\tan140^\circ=-\tan40^\circ$
Question 33 video solutions
Exact Values Using the ASTC Rule
Question
Find the exact value of each of the following:
(a) $\tan210^\circ$
(b) $\cos\frac{8\pi}{3}$
(c) $\sin330^\circ$
(a) $\tan210^\circ$
(b) $\cos\frac{8\pi}{3}$
(c) $\sin330^\circ$
Video lesson
Study Guide - Trigonometric Functions Question 3 (a)
Video lesson
Part (b)
Video lesson
Part (c)
Step-by-step solution
- 1(a) $210^\circ=180^\circ+30^\circ$. Tangent is positive in Quadrant III, so $\tan210^\circ=\tan30^\circ=\frac{\sqrt{3}}{3}$.
- 2(b) Reduce $\frac{8\pi}{3}$ by $2\pi$: $\frac{8\pi}{3}-2\pi=\frac{2\pi}{3}$. This lies in Quadrant II with basic angle $\frac{\pi}{3}$, so $\cos\frac{8\pi}{3}=-\cos\frac{\pi}{3}=-\frac{1}{2}$.
- 3(c) $330^\circ=360^\circ-30^\circ$. Sine is negative in Quadrant IV, so $\sin330^\circ=-\sin30^\circ=-\frac{1}{2}$.
Final answer
(a) $\frac{\sqrt{3}}{3}$, (b) $-\frac{1}{2}$, (c) $-\frac{1}{2}$
Question 4Video solution
Reciprocal Trigonometric Ratios
Question
Given that $\cos\theta=\frac{2}{7}$ and $\theta$ is acute, without using a calculator, find:
(i) $\sec\theta$
(ii) $\cot\theta$
(iii) $\csc\theta$.
(i) $\sec\theta$
(ii) $\cot\theta$
(iii) $\csc\theta$.
Video lesson
Study Guide - Trigonometric Functions Question 4
Step-by-step solution
- 1Represent $\cos\theta=\frac{2}{7}$ with adjacent side $2$ and hypotenuse $7$.
- 2By Pythagoras, the opposite side is $\sqrt{7^2-2^2}=\sqrt{45}=3\sqrt{5}$.
- 3(i) $\sec\theta=\frac{1}{\cos\theta}=\frac{7}{2}$.
- 4(ii) $\cot\theta=\frac{2}{3\sqrt{5}}=\frac{2\sqrt{5}}{15}$.
- 5(iii) $\csc\theta=\frac{7}{3\sqrt{5}}=\frac{7\sqrt{5}}{15}$.
Final answer
(i) $\frac{7}{2}$, (ii) $\frac{2\sqrt{5}}{15}$, (iii) $\frac{7\sqrt{5}}{15}$
Question 5Video solution
Principal Values
Question
Write down the principal value of $\tan^{-1}(-\sqrt{3})$, leaving your answer in radians as a multiple of $\pi$.
Video lesson
Study Guide - Trigonometric Functions Question 5
Step-by-step solution
- 1The reference angle satisfying $\tan\theta=\sqrt{3}$ is $60^\circ$.
- 2The principal-value range for inverse tangent is $-90^\circ<\theta<90^\circ$, so the negative value is $-60^\circ$.
- 3Convert to radians: $-60^\circ\times\frac{\pi}{180^\circ}=-\frac{\pi}{3}$.
Final answer
$-\frac{\pi}{3}$
Revision summary
Key Formulas to Remember
Pythagorean Identity
sin^2 x + cos^2 x = 1
Use this to convert between sine and cosine.
Tangent Ratio
tan x = sin x / cos x
Useful when a question gives sine and cosine.
Double-Angle Formula
sin 2x = 2 sin x cos x
Double-Angle Formula
cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x
R-Formula
a sin x + b cos x = R sin(x + alpha), where R = sqrt(a^2 + b^2)
Identities and Results to Memorise
- sin^2 x + cos^2 x = 1
- tan x = sin x / cos x
- 1 + tan^2 x = sec^2 x
- sin 2x = 2sin x cos x
- cos 2x = 1 - 2sin^2 x
- cos 2x = 2cos^2 x - 1
Need Help With A Math Trigonometry?
Our Additional Mathematics classes help Sec 3 and Sec 4 students build the algebra and trigonometry foundations needed for O-Level and H2 Math.