Question

Please help!!!!!!! (100 points!!!!)

How many solutions exist for the equation cos 2θ − sin^2 θ = 0 on the interval [0, 360°)?

3
2
1
0

Answer 1

4 solution

Explanation:

cos 2θ − sin^2 θ = 0

Use trig identity cos 2θ = 1 - 2sin² θ

cos 2θ − sin^2 θ = 0

1 - 2sin² θ − sin^2 θ = 0

-3sin² θ = -1

sin² θ = 1/3

sin θ = ±√(1/3)

The reference angle is 35.26°.

Since the interval is [0, 360°), there are 4 solutions:

35.26°, 144.74°, 215.26°, 324.74°

There are 4 solutions.

Answer 2

There lies 2 solutions.

Explanation:

`cos\:2x\:-\:sin^2\:x\:=\:0\:`

rewrite the expression:

`\cos ^2\left(x\right)-2\sin ^2\left(x\right)=0`

Factor the expression:

`\left(\cos \left(x\right)+√(2)\sin \left(x\right)\right)\left(\cos \left(x\right)-√(2)\sin \left(x\right)\right)=0`

solve them separately:

`\cos \left(x\right)+√(2)\sin \left(x\right)=0\quad \ \ or \ \ \cos \left(x\right)-√(2)\sin \left(x\right)=0`

final answer:

`35.26438^(\circ \:)` ,
`215.3^(\circ \:)`