Use these Inter 1st Year Maths 1A Formulas PDF Chapter 7 Trigonometric Equations Functions to solve questions creatively.

## Intermediate 1st Year Maths 1A Trigonometric Equations Formulas

→ If K ∈ [-1, 1], then the principal solution θ of sin θ = K lies in \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]\)and the general solution is given by nπ + (-1)^{n} θ, n ∈ Z

→ If K ∈ [-1, 1], then the principal solution θ of cos θ = K lies in [0, π] and the general solution is given by 2nπ ± 0, n ∈ Z

→ If K ∈ R, then the principal solution θ of tan x = K lies in \(\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\), and the general solution is given by n\left(\frac{-\pi}{2}, \frac{\pi}{2}\right) ± θ, n ∈ Z

→ sin θ = 0 ⇒ θ = nπ, n is an integer and its principal solution is x = 0

→ cos θ = 0 ⇒ θ = (2n + 1)\(\frac{\pi}{2}\), n is an integer and its principal solution is x = \(\frac{\pi}{2}\)

→ tan θ = 0 ⇒ θ = nπ, n is an integer and its principal solution is x = 0

→ cot θ = 0 ⇒ θ = (2n + 1)\(\frac{\pi}{2}\) n is an integer

→ cosec θ = cosec α ⇒ θ = nπ + (-1)^{n}α, n is an integer

→ sec θ = sec α ⇒ θ = 2nπ ± α, n is an integer

→ cot θ = cot α ⇒ θ = nπ ± α and the principal solution α ∈ (0, π)

→ sin^{2}θ = sin^{2} α ⇒ θ = nπ ± α, wherenever a solution exists.

→ cos^{2}θ = cos^{2} α ⇒ θ = nπ ± α, wherenever a solution exists.

→ tan^{2}θ = tan^{2} α ⇒ θ = nπ ± α, wherenever a solution exists.

→ a cos θ + b sin θ = c, (a, b, c) ∈ R and a^{2} + b^{2} ≠ 0 has a solution, iff c^{2} ≤ a^{2} + b^{2}.

Trigonometric Equations:

An equation consisting of the trigonometric functions of a variable angle θ ∈ R is called a trigonometric equation.

→ sin θ = 0 ⇒ θ = nπ, where n ∈ Z.

→ cos θ = 0 ⇒ θ = (2n + 1)\(\), where n ∈ Z.

→ tan θ = 0 ⇒ θ = nπ where π is any integer.

→ The solution of sin θ = k (|k|<1) lying between – \(\frac{\pi}{2}\) and \(\frac{\pi}{2}\) is called the principle solution of the equation.

→ The solution of cos θ = k (|k|<1) lying between 0 and π is called the principal solution of the equation.

→ That solution of tan θ = k, lying between – \(\frac{\pi}{2}\) and \(\frac{\pi}{2}\) is called the principal solution of the equation.

→ The general value of θ satisfying cos θ = k (|k| < 1) is given by θ = 2nπ ± α where n ∈Z.

→ The general value of θ satisfying sin θ = k (|k|< 1) is given by θ = nπ + (-1)^{n}α where n ∈ Z.

→ The general value of θ satisfying tan θ = k is given by θ = nπ + α where n ∈ Z. (in each of the above cases, α is the principal solution).

→ If sin0 = k, tan0 = k are given equations, then the general value of θ is given by θ = 2nπ + α where α is that solution lying between 0 and 2π.

→ The equation a cos θ + b sin θ = c will have no solution or will be inconsistent if |c| > \(\sqrt{a^{2}+b^{2}}\)

→ cos nπ = (-1)^{n} , sin nπ = 0.

→ If sin^{2}θ = sin^{2}α or cos^{2}θ = cos^{2}α or tan^{2}θ = tan^{2}a, then θ = nπ ± α; n ∈ Z.