Trigonometric Equations and In-equations
- If sin θ = sin α ⇒ θ = n π + (−1)n α where α ∈ [latex]-\frac{\pi}{2}, \frac{\pi}{2}[/latex] , n ∈ I.
- If cos θ = cos α ⇒ θ = 2 n π ± α where α ∈ [0 , π] , n ∈ I .
- If tan θ = tan α ⇒ θ = n π + α where α ∈ (\(-\frac{\pi}{2}, \frac{\pi}{2}\)) , n ∈ I
- If sin² θ = sin² α ⇒ θ = n π ± α.
- cos² θ = cos² α ⇒ θ = n π ± α.
- tan² θ = tan² α ⇒ θ = n π ± α. [ Note: α is called the principal angle]
- Types Of Trigonometric Equations:
- Solutions of equations by factorising . Consider the equation ;
(2 sin x − cos x) (1 + cos x) = sin² x ; cotx – cosx = 1 – cotx cosx - 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 - 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
- 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 - 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
- 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. - Solving equations with the use of the Boundness of the functions sin x & cos x or by making two perfect squares. Consider the equations:
\(\begin{array}{l}{\sin x\left(\cos \frac{x}{4}-2 \sin x\right)+\left(1+\sin \frac{x}{4}-2 \cos x\right) \cdot \cos x=0} \\ {\sin ^{2} x+2 \tan ^{2} x+\frac{4}{\sqrt{3}} \tan x-\sin x+\frac{11}{12}=0}\end{array}\)
- Solutions of equations by factorising . Consider the equation ;
- 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:
\(\log _{2}\left(\sin \frac{x}{2}\right)<-1 ; \sin x\left(\cos x+\frac{1}{2}\right) \leq 0 ; \sqrt{5-2 \sin 2 x} \geq 6 \sin x-1\)