Trigonometric Equations

Common Trigonometric equations and inequalities

1. If sin θ = sin α ⇒ θ = n π + (−1)ⁿ α where α ∈ \[-\frac{\pi}{2}, \frac{\pi}{2}\] , 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]

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 x & cos x 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 x . cos 2 x = y.

1. 1. Solving equations with the use of the Boundness of the functions sin x & cos x or by making two perfect squares. Consider the equations:
      Trigonometric Equestion
      \(\begin{array}{l}{\sin x\left(\cos \frac{x}{4}-2 \sin x\right)+\left(1+\sin \frac{x}{4}-2 \cos x\right) \cdot \cos x=0} \\ {\sin ^{2} x+2 \tan ^{2} x+\frac{4}{\sqrt{3}} \tan x-\sin x+\frac{11}{12}=0}\end{array}\)
2. Trigonometric Inequalities:
   There is no general rule to solve 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:
   \(\log _{2}\left(\sin \frac{x}{2}\right)<-1 ; \sin x\left(\cos x+\frac{1}{2}\right) \leq 0 ; \sqrt{5-2 \sin 2 x} \geq 6 \sin x-1\)