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)nα, 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)
→ sin2θ = sin2 α ⇒ θ = nπ ± α, wherenever a solution exists.
→ cos2θ = cos2 α ⇒ θ = nπ ± α, wherenever a solution exists.
→ tan2θ = tan2 α ⇒ θ = nπ ± α, wherenever a solution exists.
→ a cos θ + b sin θ = c, (a, b, c) ∈ R and a2 + b2 ≠ 0 has a solution, iff c2 ≤ a2 + b2.
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)nα 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 sin2θ = sin2α or cos2θ = cos2α or tan2θ = tan2a, then θ = nπ ± α; n ∈ Z.