how do i solve?
sin2ɵ = cos ɵ
and
sin2ɵsinɵ=cosɵ
To solve the equation sin(2ɵ) = cos(ɵ), you can use the trigonometric identity sin(2ɵ) = 2sin(ɵ)cos(ɵ).
1. Rewrite the equation as 2sin(ɵ)cos(ɵ) = cos(ɵ).
2. Subtract cos(ɵ) from both sides of the equation: 2sin(ɵ)cos(ɵ) - cos(ɵ) = 0.
3. Factor out the common term cos(ɵ) from the left side: cos(ɵ)(2sin(ɵ) - 1) = 0.
4. Use the zero product property: cos(ɵ) = 0 or 2sin(ɵ) - 1 = 0.
For cos(ɵ) = 0:
5. Set cos(ɵ) = 0 and solve for ɵ: ɵ = π/2 + nπ, where n is an integer.
For 2sin(ɵ) - 1 = 0:
5. Add 1 to both sides of the equation: 2sin(ɵ) = 1.
6. Divide both sides by 2: sin(ɵ) = 1/2.
7. Solve for ɵ using the inverse sine function: ɵ = arcsin(1/2) = π/6 + 2nπ or 5π/6 + 2nπ, where n is an integer.
To solve the equation sin(2ɵ)sin(ɵ) = cos(ɵ), you can similarly use trigonometric identities.
1. Rewrite the equation as 2sin(ɵ)cos(ɵ)sin(ɵ) = cos(ɵ).
2. Simplify the left side by using the trigonometric identity sin(ɵ)cos(ɵ) = 1/2sin(2ɵ): sin(2ɵ)sin(ɵ) = cos(ɵ).
3. Apply the identity to get 1/2sin(2ɵ)sin(ɵ) = cos(ɵ).
4. Multiply both sides by 2: sin(2ɵ)sin(ɵ) = 2cos(ɵ).
5. Rearrange the equation: sin(2ɵ)sin(ɵ) - 2cos(ɵ) = 0.
6. Factor out cos(ɵ) from the left side: cos(ɵ)(2sin(ɵ) - sin(2ɵ)) = 0.
7. Use the zero product property: cos(ɵ) = 0 or (2sin(ɵ) - sin(2ɵ)) = 0.
For cos(ɵ) = 0, refer to the solution steps mentioned earlier.
For 2sin(ɵ) - sin(2ɵ) = 0:
8. Subtract 2sin(ɵ) from both sides of the equation: -sin(2ɵ) = -2sin(ɵ).
9. Divide both sides by -sin(ɵ) (since sin(ɵ) cannot be zero in this case): sin(2ɵ)/sin(ɵ) = 2.
10. Use the double angle identity sin(2ɵ) = 2sin(ɵ)cos(ɵ) to rewrite the equation: 2sin(ɵ)cos(ɵ)/sin(ɵ) = 2.
11. Simplify: 2cos(ɵ) = 2.
12. Divide both sides by 2: cos(ɵ) = 1.
The solution in this case is ɵ = 0 + 2nπ or ɵ = 2π + 2nπ, where n is an integer.