Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides. They are widely used in geometry, physics, engineering, and other fields to model periodic phenomena and solve various mathematical problems. The primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Here are the details of these functions:

1. Sine (sin θ):

Definition: The sine of an angle θ in a right triangle is the ratio of the length of the side opposite θ to the length of the hypotenuse. Formula: sin θ = (Opposite Side) / (Hypotenuse) Range: -1 ≤ sin θ ≤ 1 Key Properties: Sinusoidal function with a period of 360 degrees (or 2π radians). It is an odd function, meaning sin(-θ) = -sin(θ).

2. Cosine (cos θ):

Definition: The cosine of an angle θ in a right triangle is the ratio of the length of the side adjacent to θ to the length of the hypotenuse. Formula: cos θ = (Adjacent Side) / (Hypotenuse) Range: -1 ≤ cos θ ≤ 1 Key Properties: Cosine is also a sinusoidal function with a period of 360 degrees (or 2π radians). It is an even function, meaning cos(-θ) = cos(θ).

3. Tangent (tan θ):

Definition: The tangent of an angle θ in a right triangle is the ratio of the length of the side opposite θ to the length of the side adjacent to θ. Formula: tan θ = (Opposite Side) / (Adjacent Side) Range: All real numbers Key Properties: Tangent is periodic with a period of 180 degrees (or π radians). It is not defined at angles where the adjacent side is zero (e.g., 90 degrees or π/2 radians).

4. Cosecant (csc θ):

Definition: The cosecant of an angle θ is the reciprocal of the sine of θ. Formula: csc θ = 1 / sin θ Range: csc θ ≥ 1 or csc θ ≤ -1 Key Properties: The cosecant is the inverse of the sine function.

5. Secant (sec θ):

Definition: The secant of an angle θ is the reciprocal of the cosine of θ. Formula: sec θ = 1 / cos θ Range: sec θ ≥ 1 or sec θ ≤ -1 Key Properties: The secant is the inverse of the cosine function.

6. Cotangent (cot θ):

Definition: The cotangent of an angle θ is the reciprocal of the tangent of θ. Formula: cot θ = 1 / tan θ Range: All real numbers Key Properties: The cotangent is the inverse of the tangent function. It is not defined at angles where the tangent is zero.

These trigonometric functions have various properties and relationships that are essential for solving trigonometric equations, modeling periodic phenomena (e.g., oscillations, waveforms), and analyzing angles and triangles. They are fundamental tools in mathematics and science for solving problems involving angles and triangles.