# General Solution of Some Standard Equations

## General Solution of Some Standard Equations

General solution of Equation sinθ = sinα:

sinθ = sinα

sinθ – sinα = 0,

$$\left( \because \ \sin C-\sin D=2\cos \left( \frac{C+D}{2} \right)\sin \left( \frac{C-D}{2} \right) \right)$$,

$$2\cos \left( \frac{\theta +\alpha }{2} \right)\sin \left( \frac{\theta -\alpha }{2} \right)=0$$,

$$\cos \left( \frac{\theta +\alpha }{2} \right)=0$$,

$$\left( \frac{\theta +\alpha }{2} \right)={{\cos }^{-1}}\left( 0 \right)$$,

$$\left( \frac{\theta +\alpha }{2} \right)=\left( 2m+1 \right)\frac{\pi }{2}$$,

θ + α = (2m + 1) π

θ = (2m + 1) π – α, m ϵ Z

$$\theta \text{ }=\text{ }\left( 2m\text{ }+\text{ }1 \right)\text{ }\pi +{{\left( -1 \right)}^{2m+1}}\text{ }\alpha ,\text{ }m\text{ }\epsilon \text{ }Z$$….(1)

$$\sin \left( \frac{\theta -\alpha }{2} \right)=0$$

(θ – α)/2 = sin⁻¹(0)

(θ – α)/2 = mπ

(θ – α) = 2mπ

θ  = 2mπ + α  m ϵ Z

$$\theta ~=\text{ }2m\pi \text{ }+\text{ }{{\left( -1 \right)}^{2m}}\alpha ,\ \ ~m\text{ }\epsilon \text{ }Z$$…(2)

From equation (1) and (2) combining

$$\theta =n\pi +{{\left( -1 \right)}^{n}}\alpha ,\ n\in Z$$,

Note: For general solution of the equation sinθ = k, where -1 ≤ k ≤ 1.we have $$sin\theta \text{ }=\text{ }sin(si{{n}^{-1}}k)$$ then $$\theta \text{ }=n\pi +{{\left( -1 \right)}^{n}}(si{{n}^{-1}}k),\ n\in Z$$.

Example: Solve sin³θ cosθ – cos³θsinθ = ¼

solution:

sin³θ cosθ – cos³θsinθ = ¼

4sinθ cosθ(sin²θ – cos²θ) = 1

2sin2θ(-cos2θ) = 1

2sin2θ(cos2θ) = -1

sin4θ= -1

4θ = sin⁻¹(-1)

4θ = 2nπ – π/2, n ϵ Z