General Values and Trigonometric Equations

1. The trigonometric equation is an equation involving one or more trigonometric function of a variable. The equation may be true for one or more values, but not for every value of variable.
2. General Equation: The set of all possible solutions of trigonometric equation is called general solution of the equation.
3. Principal Value: The value of trigonometric function between 0 and 2π is known as Principal value.
4. The general solution of some trigonometric equation are–
a.
i. Sinθ = 0, θ = nπ
ii. Cosθ = 0, θ = (2n+1)π/2
iii. Tanθ = 0, θ = nπ
b.
i. Sinθ = Sinα, θ = nπ + (-1)ⁿα
ii. Cosecθ = Cosecα, θ = nπ + (-1)ⁿα
iii. Cosθ = Cosα, θ = 2nπ ± α
iv. Secθ = Secα , θ = 2nπ ± α
v. Tanθ = Tanα, θ = nπ + α
vi. Cotθ = Cotα , θ = nπ + α
c.
i. Sinθ = -Sinα, θ = nπ + (-1)ⁿ(-α)
ii. Cosθ = -Cosα, θ = 2nπ + (π-α)
iii. Tanθ = -Tanα, θ = nπ + (-α)
d.
i. Sin²θ = 1, θ = nπ + π/2
ii. Cos²θ = 1, θ = nπ
e.
i. Sin²θ = Sin²α, θ = nπ ± α
ii. Cos²θ = Cos²α, θ = nπ ± α
iii. Tan²θ = Tan²α , θ = nπ ± α
f.
i. Sinθ = 1, θ = (4n + 1)π/2
ii. Sinθ = -1, θ = (4n - 1)π/2
iii. Cosθ = 1, θ = 2nπ
iv. Cosθ = -1, θ = (2n+1)π
Note :
- Any value of x which makes both L.H.S. and R.H.S. equal will be a root but L.H.S. equal will be a root but the value of x for which ∞ = ∞ will not be solution as it is indeterminate form.
- When we square a given equation for finding solution, then, after finding roots, we have to check which root satisfies the origional equation.
- General values gives infinite solution.