# 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 Sin2 θ + Sin2α is θ = nπ ± α
• For Cos2 θ = Cos2α and Tan2 θ = Tan2α 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.

### 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 θ1 + sin θ2 + .... + sin θn = n
⇒ sin θ1 = sin θ2 = .... = sin θn = 1
i) cos θ1 + cos θ2 + .... + cos θn = n
⇒ cos θ1 = cos θ2 = .... = cos θn = 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