Trigonometry: Trigonometric Functions and Identities

Trigonometry: Understanding Functions and Identities

Trigonometry is a fundamental branch of mathematics that focuses on the relationships between the sides and angles of triangles. Two crucial aspects of trigonometry are trigonometric functions and trigonometric identities. In this article, we will explore both topics in detail.

Trigonometric Functions

There are six basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions are defined relative to the sides and angles of a right triangle.

• Sine (sin): The ratio of the side opposite an angle to the length of the hypotenuse, i.e., sin A = opposite side / hypotenuse.
• Cosine (cos): The ratio of the adjacent side to the length of the hypotenuse, i.e., cos A = adjacent side / hypotenuse.
• Tangent (tan): The ratio of the side opposite an angle to the adjacent side, i.e., tan A = opposite side / adjacent side.
• Cotangent (cot): The reciprocal of the tangent function, i.e., cot A = adjacent side / opposite side.
• Secant (sec): The reciprocal of the cosine function, i.e., sec A = hypotenuse / adjacent side.
• Cosecant (csc): The reciprocal of the sine function, i.e., csc A = hypotenuse / opposite side.

These functions are essential tools in solving problems related to the angles and sides of triangles, such as calculating unknown angles and distances.

Trigonometric Identities

Trigonometric identities are mathematical statements that relate the values of different trigonometric functions for the same angle. Some common trigonometric identities include:

• Pythagorean Identity: sin²(A) + cos²(A) = 1 for any angle A.
• Reciprocal Identities: sin(-A) = -sin(A), cos(-A) = cos(A), tan(-A) = -tan(A), cot(-A) = -cot(A), sec(-A) = -csc(A), and csc(-A) = -sec(A).
• Even and Odd Properties: sin(π - A) = -sin(A) and cos(π - A) = -cos(A) for any angle A.
• Addition Formulae: sin(A + B) = sin(A)cos(B) + cos(A)sin(B), cos(A + B) = cos(A)cos(B) - sin(A)sin(B), tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), cot(A + B) = (cot(A) + cot(B)) / (1 + cot(A)cot(B)), sec(A + B) = sec(A)sec(B) / |cos(A + B)|, and csc(A + B) = csc(A)csc(B) / |sin(A + B)|.
• Subtraction Formulae: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), cos(A - B) = cos(A)cos(B) + sin(A)sin(B), tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)), cot(A - B) = (cot(A) - cot(B)) / (1 - cot(A)cot(B)), sec(A - B) = sec(A)sec(B) / |cos(A - B)|, and csc(A - B) = csc(A)csc(B) / |sin(A - B)|.

These identities are used to simplify trigonometric expressions and equations, making calculations easier.