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 cos^{2} 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 \(\frac{-\pi}{6}\)

→ 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. sin^{2}θ = sin^{2 }α |
θ = nπ ± α, n ∈ Z |

8. cos^{2}θ = cos^{2 }α |
θ = nπ ± α, n ∈ Z |

9. tan^{2}θ = tan^{2 }α |
θ = nπ ± α, n ∈ Z |