Trigonometry is the branch of mathematics concerned with the relationships between the side lengths and angles of triangles. Derived from the Greek words trigonon (triangle) and metron (measure), get more it serves as a foundational pillar for calculus, physics, engineering, astronomy, and computer graphics.
This reference guide compiles the essential definitions, identities, unit circle properties, and non-right triangle laws into a clear, study-ready structure.
1. Fundamental Right-Triangle Trigonometry
In a right-angled triangle, trigonometry defines relationships relative to a specific reference angle, usually denoted by $\theta$.
The Primary Trigonometric Ratios
Using the mnemonic SOH CAH TOA, the three fundamental ratios are defined as:
$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
$$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$$
The Reciprocal Ratios
Each primary function has an inverse ratio (reciprocal):
$$\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$
$$\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$
$$\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}}$$
2. Radians and the Unit Circle
While geometry frequently uses degrees, higher mathematics relies on radians. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
$$\pi \text{ radians} = 180^\circ$$
$$\text{Degrees to Radians: } \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180^\circ}$$
$$\text{Radians to Degrees: } \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi}$$
Defining Functions on the Unit Circle
The Unit Circle is a circle centered at the origin $(0,0)$ with a radius of $r = 1$. For any point $(x, y)$ on the unit circle corresponding to an angle $\theta$ measured counterclockwise from the positive $x$-axis:
$$\cos(\theta) = x, \quad \sin(\theta) = y, \quad \tan(\theta) = \frac{y}{x}$$
Standard Exact Values Reference Table
| Angle (θ) in Degrees | Angle (θ) in Radians | sin(θ) | cos(θ) | tan(θ) |
| $0^\circ$ | $0$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ |
| $45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $60^\circ$ | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $90^\circ$ | $\frac{\pi}{2}$ | $1$ | $0$ | Undefined |
| $180^\circ$ | $\pi$ | $0$ | $-1$ | $0$ |
| $270^\circ$ | $\frac{3\pi}{2}$ | $-1$ | $0$ | Undefined |
| $360^\circ$ | $2\pi$ | $0$ | $1$ | $0$ |
Signs in the Four Quadrants (“ASTC”)
The acronym ASTC (“All Students Take Calculus”) identifies which primary trigonometric functions are positive in each quadrant (starting from Quadrant I and moving counterclockwise):
- Quadrant I ($0$ to $\frac{\pi}{2}$): All functions are positive.
- Quadrant II ($\frac{\pi}{2}$ to $\pi$): Sine (and cosecant) are positive.
- Quadrant III ($\pi$ to $\frac{3\pi}{2}$): Tangent (and cotangent) are positive.
- Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): Cosine (and secant) are positive.
3. Essential Trigonometric Identities
Identities are equations involving trigonometric functions that hold true for every value of the occurring variables.
Fundamental Pythagorean Identities
Derived directly from the Pythagorean theorem ($a^2 + b^2 = c^2$):
$$\sin^2(\theta) + \cos^2(\theta) = 1$$
$$1 + \tan^2(\theta) = \sec^2(\theta)$$
$$1 + \cot^2(\theta) = \csc^2(\theta)$$
Even / Odd Identities (Symmetry)
- Sine (Odd Function): $\sin(-\theta) = -\sin(\theta)$
- Cosine (Even Function): $\cos(-\theta) = \cos(\theta)$
- Tangent (Odd Function): $\tan(-\theta) = -\tan(\theta)$
Co-Function Identities
$$\sin\left(\frac{\pi}{2} – \theta\right) = \cos(\theta), \quad \cos\left(\frac{\pi}{2} – \theta\right) = \sin(\theta)$$
$$\tan\left(\frac{\pi}{2} – \theta\right) = \cot(\theta), \quad \sec\left(\frac{\pi}{2} – \theta\right) = \csc(\theta)$$
4. Advanced Formulas: Compound Angles & Double Angles
Angle Sum and Difference Formulas
$$\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$$
$$\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$$
$$\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$$
Double-Angle Formulas
$$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$
$$\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta) = 2\cos^2(\theta) – 1 = 1 – 2\sin^2(\theta)$$
$$\tan(2\theta) = \frac{2\tan(\theta)}{1 – \tan^2(\theta)}$$
Power-Reduction Formulas
Useful in integral calculus to lower the degree of trigonometric powers:
$$\sin^2(\theta) = \frac{1 – \cos(2\theta)}{2}$$
$$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$$
5. Non-Right Triangle Laws (Oblique Triangles)
When solving non-right triangles where side lengths are $a, b, c$ opposite to angles $A, B, C$ respectively:
The Law of Sines
Used when you know AAS, ASA, or SSA (the ambiguous case):
$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$
The Law of Cosines
Used when you know SAS or SSS:
$$c^2 = a^2 + b^2 – 2ab\cos(C)$$
$$a^2 = b^2 + c^2 – 2bc\cos(A)$$
$$b^2 = a^2 + c^2 – 2ac\cos(B)$$
Area of an Oblique Triangle
Given two sides and the included angle:
$$\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B)$$
6. Real-World Applications
- Physics & Wave Mechanics: Simple harmonic motion, acoustic vibrations, light propagation, and AC circuits rely on sine and cosine waves defined by $y = A\sin(\omega t + \phi)$.
- Navigation & Vectors: Breaking forces, velocities, Bonuses or displacement vectors into perpendicular components ($x = R\cos(\theta)$, $y = R\sin(\theta)$).
- Computer Graphics & Game Development: Rotations in 2D and 3D space are governed by matrix transformations built on trigonometric ratios.