The Six Trigonometric Functions: Defining Relationships in Right Triangles

Trigonometric functions are mathematical functions that relate the angles of a right-angled triangle to the ratios of its side lengths. They are fundamental in geometry, physics, engineering, and many other fields for analyzing periodic phenomena like waves and oscillations. While initially defined for right triangles, they can be extended to any angle using the unit circle.

In a right triangle with hypotenuse (h), the side opposite (o) the angle θ, and the side adjacent (a) to the angle θ:

Sine (sin θ): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
sin θ = Opposite / Hypotenuse = o/h
Cosine (cos θ): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
cos θ = Adjacent / Hypotenuse = a/h
Tangent (tan θ): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. It can also be expressed as the ratio of sine to cosine.
tan θ = Opposite / Adjacent = o/a = sin θ/cos θ
Cotangent (cot θ): The reciprocal of the tangent function. It is the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.
cot θ = Adjacent / Opposite = a/o = cos θ/sin θ
Secant (sec θ): The reciprocal of the cosine function. It is the ratio of the length of the hypotenuse to the length of the side adjacent to the angle.
sec θ = Hypotenuse / Adjacent = h/a = 1/cos θ
Cosecant (csc θ): The reciprocal of the sine function. It is the ratio of the length of the hypotenuse to the length of the side opposite the angle.
csc θ = Hypotenuse / Opposite = h/o = 1/sin θ

Unit Circle and Periodicity: Extending Trig Functions Beyond Triangles

The unit circle is a powerful tool for understanding trigonometric functions for any angle, not just acute angles in right triangles. It's a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane. The coordinates (x, y) of a point on the unit circle correspond to (cos θ, sin θ) for the angle θ formed with the positive x-axis. This helps visualize the periodic nature of these functions.

Periods: The period of a trigonometric function is the length of the smallest interval over which the function's values repeat. Understanding periodicity is crucial for solving trigonometric equations and modeling cyclical phenomena.
- sin, cos: 2π (360°): Both sine and cosine functions complete one full cycle every 2π radians (or 360 degrees). This means their values repeat after every 2π interval.
- tan, cot: π (180°): Tangent and cotangent functions have a shorter period, repeating every π radians (or 180 degrees).
- sec, csc: 2π (360°): Secant and cosecant functions, being reciprocals of cosine and sine respectively, also have a period of 2π radians (or 360 degrees).
Domains and Ranges: The domain of a function is the set of all possible input values (angles), and the range is the set of all possible output values (the ratio). Understanding these helps identify where functions are defined and what values they can produce.
- sin: Domain: ℝ (all real numbers), Range: [-1,1]: The sine function can take any real angle as input, and its output value will always be between -1 and 1, inclusive.
- cos: Domain: ℝ (all real numbers), Range: [-1,1]: Similar to sine, the cosine function accepts any real angle, and its output is also bounded between -1 and 1.
- tan: Domain: x ≠ π/2 + πn (where n is an integer), Range: ℝ (all real numbers): The tangent function is undefined at angles where cosine is zero (e.g., 90°, 270°, etc.), as it involves division by zero. Its output can be any real number.
- cot: Domain: x ≠ πn (where n is an integer), Range: ℝ (all real numbers): The cotangent function is undefined where sine is zero (e.g., 0°, 180°, etc.). Its output can also be any real number.
- sec: Domain: x ≠ π/2 + πn (where n is an integer), Range: (-∞,-1]∪[1,∞): The secant function is undefined where cosine is zero. Its output values are always greater than or equal to 1 or less than or equal to -1.
- csc: Domain: x ≠ πn (where n is an integer), Range: (-∞,-1]∪[1,∞): The cosecant function is undefined where sine is zero. Its output values are also always greater than or equal to 1 or less than or equal to -1.

Advanced Properties: Symmetries and Fundamental Relationships

Beyond their basic definitions, trigonometric functions exhibit various symmetries and relationships that are crucial for simplifying expressions, solving equations, and understanding their behavior in more complex mathematical contexts.

Even/Odd Functions: Symmetry About the Y-axis or Origin

A function is even if f(-x) = f(x) (symmetric about the y-axis), and odd if f(-x) = -f(x) (symmetric about the origin). This property helps simplify expressions involving negative angles.

sin: Odd: sin(-θ) = -sin(θ). For example, sin(-30°) = -sin(30°).
cos: Even: cos(-θ) = cos(θ). For example, cos(-30°) = cos(30°).
tan: Odd: tan(-θ) = -tan(θ).
cot: Odd: cot(-θ) = -cot(θ).
sec: Even: sec(-θ) = sec(θ).
csc: Odd: csc(-θ) = -csc(θ).

Fundamental Identities: Core Relationships Between Functions

These identities are equations that are true for all values of the variables for which both sides of the equation are defined. They are derived from the definitions of the functions and the Pythagorean theorem, forming the backbone of trigonometry.

sin²θ + cos²θ = 1: The most fundamental Pythagorean identity, stating that the square of the sine of an angle plus the square of its cosine always equals 1. This comes directly from the unit circle.
1 + tan²θ = sec²θ: This identity relates tangent and secant. It can be derived by dividing the first Pythagorean identity by cos²θ.
1 + cot²θ = csc²θ: This identity connects cotangent and cosecant. It can be derived by dividing the first Pythagorean identity by sin²θ.
sin(2θ) = 2sinθcosθ: A double angle identity for sine, useful for simplifying expressions involving twice an angle.
cos(2θ) = cos²θ - sin²θ: One form of the double angle identity for cosine. Other forms include 2cos²θ - 1 and 1 - 2sin²θ.

Special Angles Table: Exact Values for Common Angles

Certain angles are frequently encountered in trigonometry and have exact, easily memorized trigonometric function values. These values are derived from special right triangles (like 30-60-90 and 45-45-90 triangles) or key points on the unit circle. Knowing these exact values can significantly speed up calculations and problem-solving in various mathematical and scientific contexts.

Angle	sin	cos	tan
0° (0 rad)	0	1	0
30° (π/6 rad)	1/2	√3/2	1/√3
45° (π/4 rad)	1/√2 (or √2/2)	1/√2 (or √2/2)	1
60° (π/3 rad)	√3/2	1/2	√3
90° (π/2 rad)	1	0	undefined

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