Two sides and the included angle.

Note: If $A=90^\circ$, $\cos A = 0$, so $a^2 = b^2 + c^2$ (Pythagoras' Theorem).

TMUA/ESAT Tip: Understand when to use each rule. Be aware of the ambiguous case for the Sine Rule and how to handle it. Clearly label diagrams.

MM4.2 Radian Measure

Definition: One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
Conversions:
- $\pi$ radians $= 180^\circ$.
- $1$ radian $\approx 57.298^\circ$.
- Degrees to radians: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$.
- Radians to degrees: $\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}$.
Standard Conversions:
- $30^\circ = \pi/6$ rad
- $45^\circ = \pi/4$ rad
- $60^\circ = \pi/3$ rad
- $90^\circ = \pi/2$ rad
- $180^\circ = \pi$ rad
- $360^\circ = 2\pi$ rad
Arc Length ($L$): $L = r\theta$ (where $\theta$ is in radians).
Area of Sector ($A_{sector}$): $A_{sector} = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians).

TMUA/ESAT Strategy: Always ensure angles are in radians when using arc length and sector area formulas. Be familiar with standard angle conversions.

MM4.3 Values of Sine, Cosine, and Tangent for Standard Angles

$0^\circ$ (0 rad): $\sin 0 = 0$, $\cos 0 = 1$, $\tan 0 = 0$.
$30^\circ$ ($\pi/6$ rad): $\sin 30^\circ = 1/2$, $\cos 30^\circ = \sqrt{3}/2$, $\tan 30^\circ = 1/\sqrt{3}$.
$45^\circ$ ($\pi/4$ rad): $\sin 45^\circ = 1/\sqrt{2}$, $\cos 45^\circ = 1/\sqrt{2}$, $\tan 45^\circ = 1$.
$60^\circ$ ($\pi/3$ rad): $\sin 60^\circ = \sqrt{3}/2$, $\cos 60^\circ = 1/2$, $\tan 60^\circ = \sqrt{3}$.
$90^\circ$ ($\pi/2$ rad): $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ$ is undefined.

TMUA/ESAT Trick: Learn these values or be able to quickly derive them using equilateral and isosceles right-angled triangles.

MM4.4 Sine, Cosine, and Tangent Functions: Graphs, Symmetries, Periodicity

Sine Function ($y = \sin x$):
- Period $2\pi$ ($360^\circ$).
- Range $[-1, 1]$.
- Odd function: $\sin(-x) = -\sin x$.
- Symmetry about $x = \frac{\pi}{2} + n\pi$.
Cosine Function ($y = \cos x$):
- Period $2\pi$ ($360^\circ$).
- Range $[-1, 1]$.
- Even function: $\cos(-x) = \cos x$.
- Symmetry about $x = n\pi$.
Tangent Function ($y = \tan x$):
- Period $\pi$ ($180^\circ$).
- Range $(-\infty, \infty)$.
- Odd function: $\tan(-x) = -\tan x$.
- Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$.
CAST Diagrams / Projections: Visualise the signs of $\sin$, $\cos$, $\tan$ in different quadrants.
- $\sin \theta$: y-coordinate projection.
- $\cos \theta$: x-coordinate projection.
- $\tan \theta$: slope of the radius.

TMUA/ESAT Strategy: Use graphing tools (like Desmos) to explore transformations and relationships between trigonometric graphs. Understand how values repeat due to periodicity.

MM4.5 Trigonometric Identities

Quotient Identity: $\tan \theta = \frac{\sin \theta}