Trigonometric Identities: Odd and Even Properties, Ratio Relationships, and Angles π, π/2, and 3π/2

1. Odd and Even Properties of Trigonometric Functions

Some trigonometric functions have odd or even properties, which are very useful for simplifying expressions.

Sine (Odd Function):

sin(-α) = -sin(α)

Cosine (Even Function):

cos(-α) = cos(α)

Tangent (Odd Function):

tan(-α) = -tan(α)

Cotangent (Odd Function):

cot(-α) = -cot(α)

2. Fundamental Relationships Between Trigonometric Ratios

Trigonometric ratios are directly related to each other through basic identities.

Tangent and Cotangent Definitions:

tan α = sin α / cos α

cot α = cos α / sin α

Reciprocal Relationships:

tan α = 1 / cot α

cot α = 1 / tan α

3. Pythagorean Trigonometric Identities

These identities are derived from the Pythagorean theorem applied to the unit circle.

sin²α + cos²α = 1

1 + tan²α = 1 / cos²α

1 + cot²α = 1 / sin²α

4. Identities Involving Angle π

The angle π radians is equivalent to 180 degrees.

sin(π - α) = sin α

cos(π - α) = -cos α

tan(π - α) = -tan α

cot(π - α) = -cot α

5. Identities Involving Angle π/2

The angle π/2 radians is equivalent to 90 degrees.

sin(π/2 - α) = cos α

cos(π/2 - α) = sin α

tan(π/2 - α) = cot α

cot(π/2 - α) = tan α

6. Identities Involving Angle 3π/2

The angle 3π/2 radians is equivalent to 270 degrees.

sin(3π/2 - α) = -cos α

cos(3π/2 - α) = -sin α

tan(3π/2 - α) = cot α

cot(3π/2 - α) = tan α

Conclusion

Trigonometric identities play a crucial role in mathematics. Mastering odd and even properties, ratio relationships, and identities involving special angles such as π, π/2, and 3π/2 is essential for solving advanced trigonometric problems efficiently.