Solve for θ from 0 to 360 degrees [sin(2θ)]²- (cosθ+sinθ)=
looks like your equation to solve is incomplete.
you can rewrite it as
sinx + cosx + cos^2(2x) = 0
clearly x= 3π/4, π, 7π/4 are some solutions.
To find them all will take some doing.
working with our equation,
-sinx = cos^2(2x) + cosx
square both sides to get
sin^2(x) = cos^4(2x) - 2cos^2(2x)cos(x) + cos^2(x)
1-cos^2(x) = (2cos^2(x)-1)^4 - 2cos(x)(2cos^2(x)-1)^2 + cos^2(x)
Or, letting u = cosx,
1-u^2 = (2u^2-1)^4-2u(2u^2-1)^2 + u^2
16u^8-32u^6-8u^5+24u^4+8u^3-6u^2-2u = 0
2u(8u^7-16u^5-4u^4+12u^3+4u^2-3u+1) = 0
Not sure how good you are at solving such polynomials. There will also be some extraneous solutions from the squaring.
oh, yes. x=3π/2 also works.
To solve the equation [sin(2θ)]² - (cosθ+sinθ) = 0 for θ from 0 to 360 degrees, we will simplify the equation and then solve for θ.
Step 1: Simplify the equation
Let's start by expanding [sin(2θ)]². Using the double angle formula for sine, we know that sin(2θ) = 2sinθcosθ.
Therefore, [sin(2θ)]² becomes (2sinθcosθ)² = 4sin²θcos²θ.
Next, the equation becomes:
4sin²θcos²θ - (cosθ+sinθ) = 0.
Step 2: Rearrange the equation
Rearranging the equation, we have:
4sin²θcos²θ - cosθ - sinθ = 0.
Step 3: Combine like terms
Let's collect the terms with sinθ and cosθ:
(4sin²θcos²θ - sinθ) - cosθ = 0.
Step 4: Factor out sinθ
Factor out sinθ:
sinθ(4cos²θ - 1) - cosθ = 0.
Step 5: Simplify further
Simplify the expression by expanding (4cos²θ - 1):
sinθ(4cos²θ - 1) - cosθ = 0
sinθ(2cosθ + 1)(2cosθ - 1) - cosθ = 0.
Step 6: Solve for θ
To find the values of θ that satisfy the equation, we need to set each factor equal to zero and solve for θ.
First factor: sinθ = 0.
This means that θ can be 0 degrees, 180 degrees, or any other angle where sinθ = 0.
Second factor: 2cosθ + 1 = 0.
Subtracting 1 from both sides, we have:
2cosθ = -1
cosθ = -1/2.
The angle θ that satisfies cosθ = -1/2 is 120 degrees or 240 degrees (or any other angle with the same cosine value).
Third factor: 2cosθ - 1 = 0.
Adding 1 to both sides, we have:
2cosθ = 1
cosθ = 1/2.
The angle θ that satisfies cosθ = 1/2 is 60 degrees or 300 degrees (or any other angle with the same cosine value).
Therefore, the values of θ from 0 to 360 degrees that satisfy the equation [sin(2θ)]²- (cosθ+sinθ) = 0 are:
θ = 0, 60, 120, 180, 240, 300 degrees.