how would you solve this equation?
294 sin^2 theta = 196 sin 2 theta
To solve this equation, we can follow these steps:
Step 1: Use trigonometric identities to simplify the equation.
Recall the identity: sin 2θ = 2sinθcosθ
We can rewrite the equation as:
294 sin^2θ = 196(2sinθcosθ)
Step 2: Expand and rearrange the equation.
Distribute 196 to the expression inside the brackets:
294 sin^2θ = 392sinθcosθ
Step 3: Rearrange the equation and factor out sinθ.
Bring all the terms to one side of the equation:
294 sin^2θ - 392sinθcosθ = 0
Factor out sinθ from both terms:
sinθ(294 sinθ - 392cosθ) = 0
Step 4: Set each factor equal to zero and solve for θ.
a) sinθ = 0
This gives us θ = 0 and θ = π (or any integer multiple of π).
b) 294 sinθ - 392cosθ = 0
Divide both sides by 294:
sinθ - (392/294)cosθ = 0
Simplify the fraction:
sinθ - (4/3)cosθ = 0
We can use the identity: sin^2θ + cos^2θ = 1
Square both sides of the equation:
(sinθ)^2 - 2(sinθ)(4/3)(cosθ) + (4/3)^2(cosθ)^2 = 0
Simplify:
(sinθ)^2 - (8/3)(sinθ)(cosθ) + (16/9)(cosθ)^2 = 0
Notice that the left side of this equation can be factored as a perfect square:
((sinθ) - (4/3)(cosθ))^2 = 0
Taking the square root of both sides:
(sinθ) - (4/3)(cosθ) = 0
Rearranging and dividing both sides by cosθ:
sinθ = (4/3)(cosθ)
Divide both sides by cosθ:
tanθ = 4/3
Take the inverse tangent of both sides to find the solution for θ:
θ = arctan(4/3)
In conclusion, the solutions to the equation 294 sin^2θ = 196 sin 2θ are: θ = 0, θ = π, and θ = arctan(4/3).