Trigonometric identities

Trigonometric Identities

  • A trigonometric identity is an equation that holds true for all values within the domain where both sides of the equality are defined. These identities can be employed to solve complex trigonometric equations and simplify expressions.

  • The simplest and most fundamental trigonometric identities are the reciprocal identities: cosec θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.

Pythagorean identities

  • A significant part of understanding trigonometric identities deals with understanding the Pythagorean identities. These are derived from the Pythagorean theorem in relation to the unit circle.

  • The three primary Pythagorean identities are: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Co-Function identities

  • Co-function identities indicate the relationship between sine, cosine, tangent, cotangent, secant, and cosecant of complementary angles.

  • Key examples of these identities are: sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, tan(90° - θ) = cot θ, cot(90° - θ) = tan θ, sec(90° - θ) = cosec θ, and cosec(90° - θ) = sec θ.

Even-Odd identities

  • The even-odd identities involve understanding which trigonometric functions are even (symmetrical around the y-axis) and which are odd (symmetrical around the origin).

  • The identities state: sin(-θ) = -sin θ, cos(-θ) = cos θ, tan(-θ) = -tan θ, cot(-θ) = -cot θ, sec(-θ) = sec θ, and cosec(-θ) = cosec θ.

Double and Half Angle identities

  • Double-angle identities are useful to simplify expressions involving the powers of sine and cosine. They include: sin 2θ = 2 sin θ cos θ, and cos 2θ = cos² θ - sin² θ.

  • Half-angle identities, which can be derived from double-angle identities, include, for example: sin θ/2 = √[(1 - cos θ) / 2] and cos θ/2 = √[(1 + cos θ) / 2].

Importance of Trigonometric Identities

  • These identities play an instrumental role in simplifying complex trigonometric equations, proving more complex identities, and solving trigonometric equations.

  • Remember, memorising these identities isn’t necessarily a goal; it is more important to understand where they come from and how they are derived. This way, even if you forget, you can still derive it rather quickly.