Understanding Trigonometry: Essential Guide for Students

Trigonometry is the branch of mathematics concerned with the relationships between angles and side lengths in triangles, with principal use of trigonometric functions and identities.

Definition and Notation of Trigonometric Ratios

Let $\triangle ABC$ be a right-angled triangle at $B$, with angle $A = \theta$. The sides are denoted as: opposite side (to $\theta$), adjacent side (next to $\theta$), and hypotenuse (longest side).

Definition: The six primary trigonometric ratios for angle $\theta$ are defined as follows.

Trigonometric Ratio	Meaning
$\sin \theta$	Opposite / Hypotenuse
$\cos \theta$	Adjacent / Hypotenuse
$\tan \theta$	Opposite / Adjacent
$\csc \theta$	Hypotenuse / Opposite
$\sec \theta$	Hypotenuse / Adjacent
$\cot \theta$	Adjacent / Opposite

Result: The trigonometric functions are defined for all real angles, with exceptions at certain non-permissible values.

Refer to Trigonometric Functions for detailed functional properties and domain restrictions.

Trigonometric Ratios for Standard Angles

The evaluation of trigonometric ratios at standard angles (such as $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$) is frequently required in competitive examinations.

Angle	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	$1$	$0$	Not defined

All values for reciprocal ratios (cosec, sec, cot) can be computed using their definitions.

Trigonometric Functions in the Cartesian Plane and Unit Circle

On the unit circle ($x^2 + y^2 = 1$), any angle $\theta$ corresponds to the point $(\cos \theta, \sin \theta)$. The sign and value of each trigonometric ratio depend on the quadrant in which $\theta$ lies.

• Trigonometric function signs by quadrant
• Values on the unit circle
• Periodic nature of functions

For a geometric interpretation, see the Trigonometric Functions page.

Fundamental Trigonometric Identities

Identity: The following results establish algebraic connections between trigonometric ratios of the same argument.

• $\sin^2 \theta + \cos^2 \theta = 1$
• $1 + \tan^2 \theta = \sec^2 \theta$
• $1 + \cot^2 \theta = \csc^2 \theta$

Additional formulas for compound, double, and half angles include:

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

For extensions and proofs, refer to Trigonometric Ratios of Compound Angles.

Degree and Radian Measure

Result: $360^\circ$ corresponds to $2\pi$ radians. The relationship between degree and radian is given by:

• $\text{Degrees} = \frac{180}{\pi} \times \text{Radians}$
• $\text{Radians} = \frac{\pi}{180} \times \text{Degrees}$

Periodicity and Reduction of Angles

Trigonometric functions exhibit periodic behavior. For any integer $k$:

• $\sin(\theta + 2k\pi) = \sin\theta$
• $\cos(\theta + 2k\pi) = \cos\theta$
• $\tan(\theta + k\pi) = \tan\theta$

Angle reduction formulas allow evaluation of functions for angles outside $[0, 2\pi)$.

Solving Triangles (Non-Right-Angled Cases)

For $\triangle ABC$ (arbitrary triangle), the Law of Sines and Law of Cosines enable computation of unknown sides and angles.

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

For practice, consult Trigonometry Practice Paper.

Worked Examples

Example: Find the value of $\sin 45^\circ$ using definition.

Given: $\theta = 45^\circ$ in a right triangle with both legs equal, so opposite = adjacent. Hypotenuse: $\sqrt{2}$ times a leg.

Substitution: $\sin 45^\circ = \frac{1}{\sqrt{2}}$

Final result: $\sin 45^\circ = \frac{1}{\sqrt{2}}$

Example: Prove $\sin^2 \theta + \cos^2 \theta = 1$ for $\theta = 30^\circ$.

Given: $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$

Substitution: $(\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 = \frac{1}{4} + \frac{3}{4} = 1$

Final result: Identity verified for $30^\circ$

Example: If $\tan \theta = 1$, find $\theta$ in $0^\circ < \theta < 90^\circ$.

Given: $\tan \theta = 1$

Solution: $\theta = 45^\circ$

Final result: $\theta = 45^\circ$

Example: If $\sec x = 2$, $0^\circ < x < 90^\circ$, find $x$.

Given: $\sec x = 2 \implies \cos x = \frac{1}{2}$

Solution: $x = 60^\circ$

Final result: $x = 60^\circ$

Common Misconceptions in Trigonometry

• Confusing quotient identities for $\tan \theta$
• Incorrect sign usage by quadrant
• Neglecting to use radian measure
• Mistaking reciprocal relations in evaluation

Revision and further notes are available at Trigonometry Revision Notes.

For related algebraic function graphing, see Graph of Quadratic Polynomial.

For competitive exam oriented questions, see Trigonometry Important Questions.