Trigonometric Identities and Their Applications

1. What Are Trigonometric Identities?

Trigonometric identities are mathematical relationships between trigonometric ratios that are true for all values of angles.

These identities are independent of specific angle values and play an essential role in simplifying expressions and solving equations.

2. Pythagorean Identities

Pythagorean identities are derived from the Pythagorean theorem and the unit circle.

sin²α + cos²α = 1

1 + tan²α = 1 / cos²α

1 + cot²α = 1 / sin²α

3. Ratio Identities

Ratio identities express trigonometric functions in terms of one another.

tan α = sin α / cos α

cot α = cos α / sin α

tan α = 1 / cot α

cot α = 1 / tan α

4. Even and Odd Identities

Some trigonometric functions exhibit even or odd behavior.

Odd Functions:

sin(-α) = -sin(α)

tan(-α) = -tan(α)

cot(-α) = -cot(α)

Even Function:

cos(-α) = cos(α)

5. Complementary Angle Identities (π/2)

These identities relate trigonometric functions of complementary angles.

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 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

7. Sum and Difference Identities

These identities are used to compute trigonometric functions of the sum or difference of two angles.

sin(α ± β) = sin α cos β ± cos α sin β

cos(α ± β) = cos α cos β ∓ sin α sin β

tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)

8. Applications of Trigonometric Identities

Trigonometric identities are widely used in simplifying expressions, solving trigonometric equations, proving mathematical relationships, and solving problems in physics and engineering.

Conclusion

Trigonometric identities are essential tools in mathematics. Mastery of Pythagorean identities, ratio identities, complementary identities, and sum and difference formulas is crucial for success in advanced trigonometry.