TRIGONOMETRY

WHY THIS MATTERS

Trigonometry is the language of angles and directions. Resolving forces into components, calculating work done by a force, expressing Simple Harmonic Motion, finding the path of projectiles — all require trigonometry. It is literally impossible to solve Class 11 mechanics without it.

§1. Angle Measurement: Degrees & Radians

Angles can be measured in two systems. Physics equations involving calculus and oscillations use Radians.

CONVERSION FORMULA

$$\pi \text{ radians} = 180°$$

$1 \text{ rad} = \frac{180°}{\pi} \approx 57.3°$ $1° = \frac{\pi}{180} \text{ rad}$

$30° = \frac{\pi}{6}$ rad

$45° = \frac{\pi}{4}$ rad

$60° = \frac{\pi}{3}$ rad

$90° = \frac{\pi}{2}$ rad

$180° = \pi$ rad

$360° = 2\pi$ rad

§2. Trigonometric Ratios (Right-Angle Triangle)

$\sin\theta = \frac{p}{h}$ (Perp/Hyp)

$\cos\theta = \frac{b}{h}$ (Base/Hyp)

$\tan\theta = \frac{p}{b} = \frac{\sin\theta}{\cos\theta}$

$\csc\theta = \frac{h}{p} = \frac{1}{\sin\theta}$

$\sec\theta = \frac{h}{b} = \frac{1}{\cos\theta}$

$\cot\theta = \frac{b}{p} = \frac{1}{\tan\theta}$

Mnemonic

"SOH-CAH-TOA": Sine = Opposite/Hypotenuse | Cosine = Adjacent/Hypotenuse | Tangent = Opposite/Adjacent

§3. Standard Trigonometric Values

Angle (θ)	0°	30°	45°	60°	90°	120°	180°	270°	360°
sin θ	0	1/2	1/√2	√3/2	1	√3/2	0	-1	0
cos θ	1	√3/2	1/√2	1/2	0	-1/2	-1	0	1
tan θ	0	1/√3	1	√3	∞	-√3	0	∞	0

Memory Trick

For sin: 0°, 30°, 45°, 60°, 90° → $\sqrt{0/4}, \sqrt{1/4}, \sqrt{2/4}, \sqrt{3/4}, \sqrt{4/4}$ = 0, ½, 1/√2, √3/2, 1. For cos, read the sin table backwards!

The 37°-53° Special Physics Triangle (3-4-5)

⭐ USED EVERYWHERE IN PHYSICS

$\sin 37° = \cos 53° = \frac{3}{5} = 0.6$

$\cos 37° = \sin 53° = \frac{4}{5} = 0.8$

$\tan 37° = \frac{3}{4} = 0.75$

$\tan 53° = \frac{4}{3} \approx 1.33$

The sides of the right triangle are in ratio 3 : 4 : 5.

§4. ASTC Rule — Signs in Quadrants

Mnemonic: A–S–T–C going from Quadrant I → II → III → IV = "All Students Take Calculus"

Allied Angle Reductions

$\sin(180° - \theta) = \sin\theta$

$\cos(180° - \theta) = -\cos\theta$

$\sin(90° + \theta) = \cos\theta$

$\cos(90° + \theta) = -\sin\theta$

$\sin(-\theta) = -\sin\theta$

$\cos(-\theta) = \cos\theta$

$\sin(270° - \theta) = -\cos\theta$

$\cos(270° - \theta) = -\sin\theta$

§5. Fundamental Identities

THREE PYTHAGOREAN IDENTITIES

$\sin^2\theta + \cos^2\theta = 1$

$1 + \tan^2\theta = \sec^2\theta$

$1 + \cot^2\theta = \csc^2\theta$

§6. Addition & Double Angle Formulas

COMPOUND ANGLES

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

DOUBLE ANGLE FORMULAS

$\sin(2\theta) = 2\sin\theta\cos\theta$

$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$

$\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$

⚛️ Physics Use — Projectile Motion: Range $R = \frac{u^2\sin 2\theta}{g}$. Maximum range at $\theta = 45°$ because $\sin 90° = 1$. This formula comes directly from the double angle identity!

§7. Small Angle Approximations

⭐ CRITICAL PHYSICS TOOL

When $\theta$ is very small (measured in radians):

$$\sin\theta \approx \theta \qquad \tan\theta \approx \theta \qquad \cos\theta \approx 1$$

Used in: Pendulum SHM, Optics (small angles), Elasticity.

Simple Pendulum Derivation:

Restoring force $F = -mg\sin\theta$. For small angles $\sin\theta \approx \theta = \frac{x}{L}$.
$\therefore F \approx -mg\frac{x}{L} = -\left(\frac{mg}{L}\right)x$. This is SHM!

§8. Graphs of Trig Functions

Graphs of sin x and cos x over [0, 2π]

Practice Questions

DRILL 1 — VALUES & CONVERSION

1. Convert 240° to radians.
2. Find the value of $\sin 150° + \cos 120°$.
3. Find $\tan 315°$.
4. If $\sin\theta = 3/5$ and $\theta$ is in Quadrant II, find $\cos\theta$ and $\tan\theta$.
5. Find the value of $\cos(-60°)$.

DRILL 2 — IDENTITIES & FORMULAS

6. Prove: $\sin^4\theta - \cos^4\theta = \sin^2\theta - \cos^2\theta$
7. Simplify: $\frac{1 - \cos^2\theta}{\sin\theta}$
8. Find $\sin 75°$ using addition formula.
9. Find $\cos 2\theta$ if $\sin\theta = 0.6$.
10. Prove: $\frac{\sin 2\theta}{1 + \cos 2\theta} = \tan\theta$

DRILL 3 — PHYSICS APPLICATIONS

11. A force of 50 N acts at 37° to the horizontal. Find its horizontal and vertical components.
12. A projectile is launched at 60° with speed 20 m/s. Find the horizontal range ($g = 10 \text{ m/s}^2$).
13. Using small angle approx, find $\sin(0.1 \text{ rad})$ approximately.
14. The angle of a pendulum is $\theta = 0.1\sin(2t)$ radians. Find the approximate restoring force if $mg = 5$ N and $L = 0.5$ m.
15. A ramp makes 53° with horizontal. A box of mass 10 kg slides on it. Find the component of gravity along and perpendicular to the ramp.

DRILL 4 — ADVANCED PROBLEMS

16. If $\tan A = 1/2$ and $\tan B = 1/3$, show that $A + B = 45°$.
17. Find all angles in $[0°, 360°]$ satisfying $2\sin^2\theta - \sin\theta - 1 = 0$.
18. Prove: $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$
19. In a triangle, $A = 60°$, $B = 75°$. Find $\sin C$ and then $\cos C$.
20. The intensity of light through a polarizer is $I = I_0\cos^2\theta$. At what angle is $I = I_0/4$?

Answer Key

Q	Answer & Method
1	$240° = \frac{4\pi}{3}$ rad
2	$\sin 150° + \cos 120° = 1/2 + (-1/2) = \mathbf{0}$
3	$\tan 315° = \tan(360°-45°) = -\tan 45° = \mathbf{-1}$
4	Q II: $\cos\theta = -4/5$, $\tan\theta = -3/4$
5	$\cos(-60°) = \cos 60° = \mathbf{1/2}$
7	$\frac{\sin^2\theta}{\sin\theta} = \mathbf{\sin\theta}$
8	$\sin 75° = \sin(45°+30°) = \frac{\sqrt{6}+\sqrt{2}}{4} \approx \mathbf{0.966}$
9	$\cos 2\theta = 1 - 2\sin^2\theta = 1 - 2(0.36) = \mathbf{0.28}$
11	$F_x = 50\cos 37° = 50(0.8) = \mathbf{40\text{ N}}$, $F_y = 50\sin 37° = 50(0.6) = \mathbf{30\text{ N}}$
12	$R = \frac{400 \sin 120°}{10} = 40 \times \frac{\sqrt{3}}{2} \approx \mathbf{34.6\text{ m}}$
13	$\sin(0.1) \approx \mathbf{0.1}$ rad
15	Along ramp: $mg\sin 53° = 10(10)(0.8) = \mathbf{80\text{ N}}$, Perpendicular: $mg\cos 53° = 10(10)(0.6) = \mathbf{60\text{ N}}$
16	$\tan(A+B) = \frac{1/2+1/3}{1-1/6} = \frac{5/6}{5/6} = 1 \implies A+B = 45°$
17	$(2\sin\theta+1)(\sin\theta-1)=0 \implies \theta = \mathbf{90°, 210°, 330°}$
19	$C = 180°-60°-75°=45°$. $\sin C = 1/\sqrt{2}$, $\cos C = 1/\sqrt{2}$
20	$\cos^2\theta = 1/4 \implies \cos\theta = 1/2 \implies \theta = \mathbf{60°}$