# Trigonometric Equations

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11 ## Trigonometric Equations and In-equations

1. If sin θ = sin α ⇒ θ = n π + (−1)n α where α ∈ [latex]-frac{pi}{2}, frac{pi}{2}[/latex] , n ∈ I.
2. If cos θ = cos α ⇒ θ = 2 n π ± α where α ∈ [0 , π] , n ∈ I .
3. If tan θ = tan α ⇒ θ = n π + α where α ∈ ((-frac{pi}{2}, frac{pi}{2})) , n ∈ I
4. If sin² θ = sin² α ⇒ θ = n π ± α.
5. cos² θ = cos² α ⇒ θ = n π ± α.
6. tan² θ = tan² α ⇒ θ = n π ± α. [ Note: α is called the principal angle]
7. Types Of Trigonometric Equations:
1. Solutions of equations by factorising . Consider the equation ;
(2 sin x − cos x) (1 + cos x) = sin² x ; cotx – cosx = 1 – cotx cosx
2. Solutions of equations reducible to quadratic equations. Consider the equation
3 cos² x − 10 cos x + 3 = 0 and 2 sin2x + (sqrt {3}) sinx + 1 = 0
3. Solving equations by introducing an Auxilliary argument. Consider the equation: sin x + cos x = (sqrt{2} ; sqrt{3}) cos x + sin x = 2 ; secx – 1 = ((sqrt {2})-1) tanx
4. Solving equations by Transforming a sum of Trigonometric functions into a product. Consider the example : cos 3 x + sin 2 x − sin 4 x = 0 ;
sin²x + sin²2x + sin²3x + sin²4x = 2 ; sinx + sin5x = sin2x + sin4x
5. Solving equations by transforming a product of trigonometric functions into a sum. Consider the equation : sin 5 x . cos 3 x = sin 6x .cos 2x ; 8cosx cos2x cos4x = (frac{sin 6 x}{sin x}) ; sin3θ = 4sinθ sin2θ sin4θ
6. Solving equations by a change of variable:
(i) Equations of the form of a . sin x + b . cos x + d = 0 , where a , b & d are real numbers & a , b ≠ 0 can be solved by changing sin & cos into their corresponding tangent of half the angle. Consider the equation 3 cos x + 4 sin x = 5.
(ii) Many equations can be solved by introducing a new variable . eg. the equation sin4 2x + cos4 2x = sin 2x . cos 2x changes to 2 (y + 1) (y − (frac {1}{2})) = 0 by substituting , sin 2 . cos 2 = y.
7. Solving equations with the use of the Boundness of the functions sin x & cos x or by making two perfect squares. Consider the equations: (begin{array}{l}{sin xleft(cos frac{x}{4}-2 sin xright)+left(1+sin frac{x}{4}-2 cos xright) cdot cos x=0} \ {sin ^{2} x+2 tan ^{2} x+frac{4}{sqrt{3}} tan x-sin x+frac{11}{12}=0}end{array})
8. Trigonometric Inequalities:
There is no general rule to solve a Trigonometric inequations and the same rules of algebra are valid except the domain and range of trigonometric functions should be kept in mind.
Consider the examples:
(log _{2}left(sin frac{x}{2}right)