## Trigonometric Functions

# Trigonometric Equations

- Equations involving trigonometric functions of a variable are called
*trigonometric equations*. - The solutions of a trigonometric equation for which 0 ≤ x ≤ 2π are called
*principal solutions.* - The expression involving integer 'n' which gives all solutions of a trigonometric equation is called the
*general solution*. **Tips:**- The general solution of the equation is Sin θ = −1 is \theta=2n\pi-\frac{\pi}{2}
- The general solution of the equation Cos θ = −1 is θ = (2n + 1)π
- The general solution of the equation Tan θ = −1 is \theta=n\pi-\frac{\pi}{4}
- The general solution of the equation Sin
^{2}θ + Sin^{2}α is θ = nπ ± α - For Cos
^{2}θ = Cos^{2}α and Tan^{2}θ = Tan^{2}α is also same. - To get the solution of the equation a cos θ + b sin θ = c. Check that |c| \leq\sqrt{a^{2}+b^{2}}. If it is not satisfied no real solution exists.
**Tricks:**- Squaring should be avoided as far as possible. If squaring is done, then check for extra solutions.
- Never cancel a common factor containing 'θ' from the two sides of an equation.
- Instead of dividing an equation by a common factor, take the factor as common factor from all terms of the equation.
- Make sure that the answer should not contain any value of unknown 'θ' which makes any of the terms undefined.
- If tanθ (or) secθ is involved in the equation, θ should not be an odd multiple of π/2.
- If cotθ (or) cosecθ is involved in the equation, θ should not be the multiple of π or 0.
- The value of \sqrt{f(\theta)} is always positive.

### Part1: View the Topic in this video From 00:40 To 55:40

### Part2: View the Topic in this video From 00:40 To 58:08

### Part3: View the Topic in this video From 00:40 To 54:46

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1. **Trigonometric Equations**

a) sin nπ = 0 and cos nπ = (−1)^{n}

b) sin θ = sin α ⇒ θ = nπ + (−1)^{n} α, n ∈ l

c) cos θ = cos α ⇒ θ = 2nπ ± α, n ∈ l

d) tan θ = tan θ ⇒ nπ + α, n ∈ l

e) sin (nπ + θ) = (−1)^{n} sin θ and cos (nπ + θ) = (−1)^{n} cos θ

f) \tt \sin \left(\frac{n \pi}{2}+\theta \right) = \left(-1\right)^{\frac{n}{2}} \cos \theta, if \ n \ is \ odd. = \left(-1\right)^{\frac{n}{2}} \sin \theta, if n is even.

g) \tt \cos \left(\frac{n \pi}{2}+ \theta \right)=\left(-1\right)^{\frac{n-1}{2}} \sin \theta, if \ n \ is \ odd.=\left(-1\right)^{\frac{n}{2}} \cos \theta, if n is even.

h) sin θ_{1} + sin θ_{2} + .... + sin θ_{n} = n

⇒ sin θ_{1} = sin θ_{2} = .... = sin θ_{n} = 1

i) cos θ_{1} + cos θ_{2} + .... + cos θ_{n} = n

⇒ cos θ_{1} = cos θ_{2} = .... = cos θ_{n} = 1

j) sin θ + cosec θ = 2 ⇒ sin θ = 1

k) cos θ + sec θ = 2 ⇒ cos θ = 1

l) sin θ + cosec θ = −2 ⇒ sin θ = −1

m) cos θ + sec θ = −2 ⇒ cos θ = −1