The biggest and highest-weightage chapter: master radians, the unit circle, ASTC quadrant rules, and all 25+ essential identities.
One radian ($1\text{ rad}$) is the angle subtended at the center of a circle of radius $r$ by an arc of length equal to $r$.
$$l = r \cdot \theta \quad (\text{where } \theta \text{ MUST be in radians!})$$
A full circle of $360^\circ$ corresponds to a circumference of $2\pi r$. Hence:
$$2\pi \text{ radians} = 360^\circ \iff \pi \text{ radians} = 180^\circ$$
| Degree | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|---|---|---|
| Radian | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ |
Consider a circle of radius $r = 1$ centered at origin $(0,0)$. For any angle $x$, the terminal ray hits the circle at point $P(a, b)$:
$$\cos x = a \quad (\text{the x-coordinate}), \qquad \sin x = b \quad (\text{the y-coordinate})$$
Since $x^2 + y^2 = 1$ on the circle, we immediately get the fundamental identity: $\cos^2 x + \sin^2 x = 1$.
Both $x > 0$ and $y > 0$.
ALL trigonometric functions are POSITIVE ($+$, $+$, $+$, $+$, $+$, $+$).
$x < 0$, $y > 0$.
Only SINE and COSECANT are POSITIVE. All others are negative.
$x < 0$, $y < 0$.
Only TANGENT and COTANGENT are POSITIVE. All others are negative.
$x > 0$, $y < 0$.
Only COSINE and SECANT are POSITIVE. All others are negative.
$$Q_1: \mathbf{A}\text{ll} \quad \longrightarrow \quad Q_2: \mathbf{S}\text{in} \quad \longrightarrow \quad Q_3: \mathbf{T}\text{an} \quad \longrightarrow \quad Q_4: \mathbf{C}\text{os}$$
Negative Angles Rule:
$$\cos(-x) = \cos x, \quad \sec(-x) = \sec x \quad (\text{Even functions — eat the minus!})$$
$$\sin(-x) = -\sin x, \quad \tan(-x) = -\tan x, \quad \cot(-x) = -\cot x, \quad \csc(-x) = -\csc x \quad (\text{Odd functions})$$
| Function | Domain | Range | Period |
|---|---|---|---|
| $\sin x$ | $\mathbb{R}$ | $[-1, 1]$ | $2\pi$ |
| $\cos x$ | $\mathbb{R}$ | $[-1, 1]$ | $2\pi$ |
| $\tan x$ | $\mathbb{R} \setminus \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$ | $\mathbb{R}$ | $\pi$ |
| $\cot x$ | $\mathbb{R} \setminus \{n\pi : n \in \mathbb{Z}\}$ | $\mathbb{R}$ | $\pi$ |
| $\sec x$ | $\mathbb{R} \setminus \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ |
| $\csc x$ | $\mathbb{R} \setminus \{n\pi : n \in \mathbb{Z}\}$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ |
1. Cosine of Sum & Difference:
$$\cos(x + y) = \cos x \cos y - \sin x \sin y$$
$$\cos(x - y) = \cos x \cos y + \sin x \sin y$$
2. Sine of Sum & Difference:
$$\sin(x + y) = \sin x \cos y + \cos x \sin y$$
$$\sin(x - y) = \sin x \cos y - \cos x \sin y$$
3. Tangent of Sum & Difference:
$$\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$$
$$\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$$
4. Cotangent:
$$\cot(x + y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}, \qquad \cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$$
1. Cosine has 4 distinct forms:
$$\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \frac{1 - \tan^2 x}{1 + \tan^2 x}$$
Power-Reduction (Lifesavers in proofs and calculus!):
$$\sin^2 x = \frac{1 - \cos 2x}{2}, \qquad \cos^2 x = \frac{1 + \cos 2x}{2}$$
2. Sine:
$$\sin 2x = 2\sin x \cos x = \frac{2\tan x}{1 + \tan^2 x}$$
3. Tangent:
$$\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$$
$$\sin 3x = 3\sin x - 4\sin^3 x$$
$$\cos 3x = 4\cos^3 x - 3\cos x$$
$$\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$$
$$2\sin x \cos y = \sin(x + y) + \sin(x - y)$$
$$2\cos x \sin y = \sin(x + y) - \sin(x - y)$$
$$2\cos x \cos y = \cos(x + y) + \cos(x - y)$$
$$-2\sin x \sin y = \cos(x + y) - \cos(x - y)$$
$$\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}$$
$$\sin C - \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2}$$
$$\cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}$$
$$\cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} = 2\sin\frac{C+D}{2}\sin\frac{D-C}{2}$$
Convert angles to radians: $\theta_1 = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3}$, $\theta_2 = 75^\circ \times \frac{\pi}{180} = \frac{5\pi}{12}$.
Given arc lengths are equal: $l_1 = l_2 \implies r_1 \theta_1 = r_2 \theta_2$.
$$\frac{r_1}{r_2} = \frac{\theta_2}{\theta_1} = \frac{5\pi/12}{\pi/3} = \frac{5}{12} \times \frac{3}{1} = \frac{5}{4}$$
Ratio of radii $r_1 : r_2 = 5 : 4$.
Using $\csc(-x) = -\csc(x)$: $\csc(-1410^\circ) = -\csc(1410^\circ)$.
Notice that $4 \times 360^\circ = 1440^\circ$.
So $1410^\circ = 4 \times 360^\circ - 30^\circ$.
Since $\csc(2n\pi - \theta) = -\csc \theta$, we have:
$$\csc(1410^\circ) = \csc(1440^\circ - 30^\circ) = -\csc 30^\circ = -2$$
$$\csc(-1410^\circ) = -(-2) = 2$$
Pair terms with symmetric angles $(4x, 2x)$ so that their average is $3x$:
Numerator: $(\cos 4x + \cos 2x) + \cos 3x = 2\cos 3x \cos x + \cos 3x = \cos 3x(2\cos x + 1)$.
Denominator: $(\sin 4x + \sin 2x) + \sin 3x = 2\sin 3x \cos x + \sin 3x = \sin 3x(2\cos x + 1)$.
Dividing numerator by denominator: $\frac{\cos 3x(2\cos x + 1)}{\sin 3x(2\cos x + 1)} = \frac{\cos 3x}{\sin 3x} = \cot 3x$.
If a question says radius $r = 10\text{ cm}$ and angle $\theta = 30^\circ$, NEVER calculate $l = 10 \times 30 = 300\text{ cm}$! You must convert $30^\circ$ to radians first: $\frac{\pi}{6}$, giving $l = \frac{10\pi}{6} = \frac{5\pi}{3}\text{ cm}$.
The identity has a minus sign: $\cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2}$. If you write positive $2$, you lose marks! Alternatively, write $2\sin\frac{C+D}{2}\sin\frac{D-C}{2}$.
$\cos(-x) = \cos x$ (absorbs minus), whereas $\sin(-x) = -\sin x$. Many students erroneously write $\cos(-\pi/3) = -\cos(\pi/3)$, which leads to sign errors.
Q1: What is the degree measure corresponding to $-\frac{2\pi}{3}$ radians?
Q2: If $\tan x = \frac{3}{4}$ and $x$ is in the 3rd quadrant, what is the value of $\sin x$?
Q3: What is the maximum value of $f(x) = 3\sin x + 4\cos x$?
Q4: What is $\cos 2x$ in terms of $\tan x$?
Q5: What is the value of $\sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ$?