Trigonometric Functions: Definitions, Formulas, Graphs, and Applications

Trigonometric functions are among the most fundamental and widely used functions in mathematics. They establish the crucial relationship between angles and side lengths, playing an indispensable role in fields ranging from high school math and university calculus to physics, engineering, and computer graphics.

This article provides a systematic overview of trigonometric functions. If you need to quickly compute trigonometric values, try our Trigonometric Calculator, which supports precise calculations for sine, cosine, tangent, and more.

1. Origins and Significance

Trigonometric functions originated from ancient astronomy and surveying. The Greek mathematician Hipparchus (c. 190–120 BC) is widely regarded as the “father of trigonometry” — he compiled the earliest known chord table in the 2nd century BC. Later, Indian mathematicians introduced the concept of sine, and Arab scholars further developed tangent and cotangent theories, eventually forming the complete trigonometric function system we use today.

The core significance of trigonometric functions lies in their ability to study geometric relationships between angles and distances using algebraic methods. This empowers us to calculate building heights, measure celestial distances, analyze wave signals, and much more.

2. Fundamental Concepts: Degrees and Radians

Before diving into trigonometric functions, it’s essential to understand the two ways of measuring angles.

2.1 Degrees

The degree is the most common unit for measuring angles. A full circle is divided into 360°.

• Right angle = 90°
• Straight angle = 180°
• Full rotation = 360°

2.2 Radians

The radian is an alternative unit of angular measurement, preferred in mathematics and physics. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius.

A full circle corresponds to 2π radians, so:

$360° = 2\pi \text{ rad} \quad \Rightarrow \quad 1° = \frac{\pi}{180} \text{ rad} \quad \Rightarrow \quad 1 \text{ rad} = \frac{180°}{\pi} \approx 57.2958°$

2.3 Common Angle Conversions

Degrees	Radians
0°	0
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
270°	3π/2
360°	2π

3. The Six Trigonometric Functions

There are six trigonometric functions, which can be defined using a right triangle or the unit circle.

3.1 Right Triangle Definition

In a right triangle, let θ be an acute angle. The side opposite to θ is a (opposite), the side adjacent to θ is b (adjacent), and the longest side is c (hypotenuse). Then:

Function	Definition	Formula
Sine (sin)	Opposite / Hypotenuse	sin θ = a / c
Cosine (cos)	Adjacent / Hypotenuse	cos θ = b / c
Tangent (tan)	Opposite / Adjacent	tan θ = a / b
Cotangent (cot)	Adjacent / Opposite	cot θ = b / a
Secant (sec)	Hypotenuse / Adjacent	sec θ = c / b
Cosecant (csc)	Hypotenuse / Opposite	csc θ = c / a

A popular mnemonic for the primary three is SOH-CAH-TOA:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

This definition is intuitive but only applies to acute angles (0° < θ < 90°).

3.2 Unit Circle Definition

To extend trigonometric functions to any angle, we use the unit circle (a circle with radius 1):

In a Cartesian coordinate system, draw a circle of radius 1 centered at the origin. Let the terminal side of angle θ intersect the unit circle at point P(x, y). Then:

$\sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x}$

$\cot\theta = \frac{x}{y}, \quad \sec\theta = \frac{1}{x}, \quad \csc\theta = \frac{1}{y}$

The advantage of the unit circle definition is that θ can be any real number (including negative angles and angles greater than 360°), making trigonometric functions defined over the entire set of real numbers.

4. Trigonometric Values of Special Angles

The following special angle values are frequently used and worth memorizing:

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	√3/3	1	√3	undefined
cot θ	undefined	√3	1	√3/3	0
sec θ	1	2√3/3	√2	2	undefined
csc θ	undefined	2	√2	2√3/3	1

💡 Memory Tip: For sine values from 0° to 90°, think of √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, 1/2, √2/2, √3/2, 1. Cosine values are the reverse.

Use our Trigonometric Calculator to verify these special values or compute the trigonometric functions of any angle.

5. Properties of Trigonometric Functions

5.1 Domain and Range

Function	Domain	Range
sin θ	All real numbers	[-1, 1]
cos θ	All real numbers	[-1, 1]
tan θ	θ ≠ kπ + π/2 (k∈ℤ)	All real numbers
cot θ	θ ≠ kπ (k∈ℤ)	All real numbers
sec θ	θ ≠ kπ + π/2 (k∈ℤ)	(-∞, -1] ∪ [1, +∞)
csc θ	θ ≠ kπ (k∈ℤ)	(-∞, -1] ∪ [1, +∞)

5.2 Periodicity

All trigonometric functions are periodic:

• sin θ and cos θ have a fundamental period of 2π (360°)
• tan θ and cot θ have a fundamental period of π (180°)
• sec θ and csc θ have a fundamental period of 2π (360°)

5.3 Symmetry (Odd and Even Functions)

• Odd functions (symmetric about the origin): sin(-θ) = -sin θ, tan(-θ) = -tan θ, cot(-θ) = -cot θ, csc(-θ) = -csc θ
• Even functions (symmetric about the y-axis): cos(-θ) = cos θ, sec(-θ) = sec θ

5.4 Signs in Each Quadrant

The signs of trigonometric functions in the four quadrants follow a specific pattern. A helpful mnemonic is “All Students Take Calculus”:

Quadrant	sin	cos	tan	cot	sec	csc
I (All)	+	+	+	+	+	+
II (Sin)	+	-	-	-	-	+
III (Tan)	-	-	+	+	-	-
IV (Cos)	-	+	-	-	+	-

6. Core Trigonometric Identities

Trigonometric identities are inherent mathematical relationships between trigonometric functions, serving as the foundation for simplification and derivation.

6.1 Reciprocal Identities

$\sin\theta \cdot \csc\theta = 1, \quad \cos\theta \cdot \sec\theta = 1, \quad \tan\theta \cdot \cot\theta = 1$

6.2 Quotient Identities

$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$

6.3 Pythagorean Identities

These are among the most important identities in trigonometry, derived directly from the Pythagorean theorem:

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

From this, we can also derive:

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

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

7. Essential Trigonometric Formulas

7.1 Sum and Difference Formulas

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$

7.2 Double Angle Formulas

$\sin 2\alpha = 2\sin\alpha\cos\alpha$

$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha$

$\tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha}$

7.3 Half Angle Formulas

$\sin\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{2}}$

$\cos\frac{\alpha}{2} = \pm\sqrt{\frac{1 + \cos\alpha}{2}}$

$\tan\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{1 + \cos\alpha}} = \frac{\sin\alpha}{1 + \cos\alpha} = \frac{1 - \cos\alpha}{\sin\alpha}$

7.4 Sum-to-Product Formulas

$\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$

$\sin\alpha - \sin\beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$

$\cos\alpha + \cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$

$\cos\alpha - \cos\beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$

7.5 Product-to-Sum Formulas

$\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]$

$\cos\alpha\sin\beta = \frac{1}{2}[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$

$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha+\beta) + \cos(\alpha-\beta)]$

$\sin\alpha\sin\beta = -\frac{1}{2}[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$

8. Graphs of Trigonometric Functions

Understanding the graphical characteristics of trigonometric functions helps build intuition about their behavior.

8.1 Sine Function: y = sin x

• Shape: A smooth, undulating wave (sinusoidal curve)
• Period: 2π
• Amplitude: 1
• Zeros: x = kπ (k∈ℤ)
• Maximum: 1, at x = π/2 + 2kπ
• Minimum: -1, at x = -π/2 + 2kπ

8.2 Cosine Function: y = cos x

• Shape: Identical to sine, but shifted left by π/2
• Period: 2π
• Amplitude: 1
• Zeros: x = π/2 + kπ (k∈ℤ)
• Maximum: 1, at x = 2kπ
• Minimum: -1, at x = π + 2kπ

8.3 Tangent Function: y = tan x

• Shape: Increases from -∞ to +∞ within each period
• Period: π
• Range: All real numbers
• Vertical Asymptotes: x = π/2 + kπ (k∈ℤ)
• Zeros: x = kπ (k∈ℤ)

8.4 General Sinusoidal Function: y = A sin(ωx + φ) + k

In practical applications, the general form of the sinusoidal function is frequently used:

• A: Amplitude — determines the height of the wave
• ω: Angular frequency — determines the density of the wave (Period T = 2π/|ω|)
• φ: Phase shift — determines the horizontal displacement
• k: Vertical shift

9. Important Theorems

9.1 Law of Sines

In any triangle ABC, where a, b, c are the sides opposite to angles A, B, C respectively, and R is the circumradius:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

Use cases: Solving triangles when two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA) are known.

9.2 Law of Cosines

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

Use cases: Solving triangles when three sides (SSS) or two sides and the included angle (SAS) are known. The Law of Cosines is also a generalization of the Pythagorean theorem — when C = 90°, cos C = 0, and the formula reduces to c² = a² + b².

10. Real-World Applications

Trigonometric functions are far more than textbook concepts — they have incredibly wide-ranging real-world applications.

10.1 Physics

• Simple Harmonic Motion: Spring oscillations and pendulum movements can be described using sine functions, e.g., $x(t) = A\sin(\omega t + \varphi)$
• Wave Theory: Sound waves, light waves, and electromagnetic waves all follow trigonometric patterns
• Alternating Current (AC): Instantaneous voltage and current values are expressed as $u = U_m\sin(\omega t + \varphi)$

10.2 Engineering

• Architectural Design: Calculating slope angles, roof pitch, and structural force analysis
• Signal Processing: Fourier transforms decompose complex signals into sine waves of different frequencies
• Navigation: GPS systems use triangulation principles to determine position

10.3 Computer Science

• Computer Graphics: 3D rotation transformations and lighting calculations rely heavily on trigonometric functions
• Game Development: Character movement directions and physics collision detection
• Animation: Using sine functions to create natural oscillating and wave effects

10.4 Everyday Life

• Height Measurement: Calculating the height of buildings or mountains using angle of elevation and distance
• Maritime & Aviation Navigation: Using trigonometric functions to compute headings and distances
• Music: Pitch frequencies and harmonic relationships are fundamentally trigonometric in nature

11. Tips for Learning Trigonometry

1. Memorize special values: The trigonometric values of 0°, 30°, 45°, 60°, and 90° are absolutely essential.
2. Master the unit circle: The unit circle is the key to understanding trigonometric functions. Draw it by hand to reinforce your understanding.
3. Learn the fundamental identities: The Pythagorean identity and quotient identities are the building blocks for deriving everything else.
4. Practice regularly: Solve computation and proof problems to solidify your understanding of formulas.
5. Use tools: Leverage our Trigonometric Calculator to verify your calculations and improve efficiency.

Conclusion

Trigonometric functions are an indispensable cornerstone of mathematics, elegantly connecting angles and lengths, and providing us with powerful mathematical tools to describe and analyze periodic phenomena in nature. From simple right triangles to complex wave equations, from ancient astronomical measurements to modern artificial intelligence, the presence of trigonometric functions is ubiquitous.

We hope this article has helped you build a systematic understanding of trigonometric functions. Whenever you need to perform trigonometric calculations for study or work, feel free to use our Trigonometric Calculator for quick and accurate results.