Question

Cos 4θ − cos 2θ = sin θ

Answer 1

To solve the equation cos 4θ - cos 2θ = sin θ, use trigonometric identities and properties. Rewrite the equation using double angle identities and the Pythagorean identity. Then, simplify the equation and solve for cos 4θ.

Step-by-step explanation:

To solve the equation cos 4θ - cos 2θ = sin θ, we can use trigonometric identities and properties. First, let's rewrite the equation using double angle identities and the Pythagorean identity. We know that cos 2θ = 1 - sin^2(θ).

Substituting this into our equation, we have cos 4θ - (1 - sin^2(2θ)) = sin θ. Expanding and simplifying, we get cos 4θ + sin^2(2θ) = 1 - sin θ.

Now, using the double angle formula, we can rewrite sin^2(2θ) as (1 - cos(4θ))/2. Substituting this into the equation, we have cos 4θ + (1 - cos(4θ))/2 = 1 - sin θ. Solving for cos 4θ and simplifying, we find cos 4θ = 1 - 2sin θ.

Answer 2

`cos(4\theta) - cos(2\theta) = sin(\theta)`

`-2sin((4\theta + 2\theta)/(2))sin((4\theta - 2\theta)/(2)) = sin(\theta)`

`-2sin((6\theta)/(2)sin(2\theta)/(2)) = sin(\theta)`

`-2sin(3\theta)sin(\theta) = sin(\theta)`

`(-2sin(3\theta)sin(\theta))/(-2sin(\theta)) = (sin(\theta))/(-2sin(\theta))`

`sin(3\theta) = -(1)/(2)`

`sin^(-1)[sin(3\theta)] = sin^(-1)(-(1)/(2))`

`3\theta = -30`

`(3\theta)/(3) = (-30)/(3)`

`\theta = -10`