TS Inter 1st Year Maths 1A Inverse Trigonometric Equations Formulas

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Learning these TS Inter 1st Year Maths 1A Formulas Chapter 7 Trigonometric Equations will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1A Trigonometric Equations Formulas

→ Trigonometric equation :
An equation involving trigonometric functions is called a trigonometric equation.
Ex – 1 : a cos2 θ – b sin θ + c = 0
Ex – 2 : a cos θ + b sin θ + c = 0
Ex – 3 : a tan θ + b sec θ + c = 0

→ General solution (or) solution set:
The set of all values of θ which satisfy a trigonometric equation f(θ) = 0 is called general solution (or) solution set of f(θ) = 0.

→ Principle value:
1. There exists a unique value of θ in \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]\) satisfying sin θ = k, k ∈ R. |k| ≤ 1. This value of θ is called principle value of θ (or) principle solution of sin θ = k.
Ex :

  • Principle solution of sin θ = \(\frac{1}{2}\) is \(\frac{\pi}{6}\).
  • Principle solution of sin θ = \(\frac{-1}{\sqrt{2}}\) is \(\frac{-\pi}{4}\).

2. There exists a unique value of θ in [0, n] satisfying cos θ = k. k ∈ R. |k| < 1. This value of θ is called principle value of θ (or) principle solution of cos θ = k.
Ex :

  • Principle solution of cos θ = \(\frac{1}{\sqrt{2}}\) is \(\frac{\pi}{4}\).
  • Principle solution of cos θ = \(\frac{-1}{2}\) is \(\frac{2 \pi}{3}\).

3. There exists a unique value of θ in \(\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\) satisfying tan θ = k, k ∈ R. This value of θ is called principle value of θ (or) principle solution of tan θ = k.
Ex :

  • Principle solution of tan θ = √3 is \(\frac{\pi}{3}\).
  • Principle solution of tan θ = \(\frac{-1}{\sqrt{3}}\) is \(\frac{-\pi}{6}\)

TS Inter 1st Year Maths 1A Trigonometric Equations Formulas

→ General solutions of trigonometric equations:

Trigonometric equation General solution
1. sin θ = 0 θ = n π, n ∈ Z
2. cos θ = 0 θ = (2n + 1) \(\frac{\pi}{2}\) , n ∈ Z
3. tan θ = 0 θ = nπ, n ∈ Z
4. sin θ = sin α θ = nπ + (- 1)n α, n ∈ Z
5. cos θ = cos α θ = 2nπ ± α, n ∈ Z
6. tan θ = tan α θ = nπ + α, n ∈ Z
7. sin2θ = sin2 α θ = nπ ± α, n ∈ Z
8. cos2θ = cos2 α θ = nπ ± α, n ∈ Z
9. tan2θ = tan2 α θ = nπ ± α, n ∈ Z

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