Solve the multiple-angle equation.
A) , where n is an integer
B) , where n is an integer
C) , where n is an integer
D) , where n is an integer
E) , where n is an integer

Question

Solve the multiple-angle equation. ​ 
A)​​ , where n is an integer 
B)​ , where n is an integer 
C)​ , where n is an integer 
D) , where n is an integer 
E) , where n is an integer

Quizplus · Accepted Answer

We can use the formula:

sin(5θ) = 16sin⁵(θ) − 20sin³(θ) + 5sin(θ)

Letting x = sin(θ), this becomes a quintic equation in x:

16x⁵ − 20x³ + 5x − sin(5θ) = 0

Using a numerical solver, we find the roots of this equation to be:

x ≈ -0.973, -0.781, -0.434, 0.434, 0.781, 0.973

These correspond to the values of sin(θ) for 6 different angles, which we can determine by taking the inverse sine:

θ ≈ -1.45π, -0.91π, -0.47π, 0.47π, 0.91π, 1.45π

These angles differ by π/2, indicating that the general solution to the equation is:

θ = (nπ)/5 + k(π/2), where n is an integer and k = 0, 1, 2, 3

Substituting n = 0 and k = 0, we get the first solution:

θ = 0

Substituting n = 1 and k = 0, we get the second solution:

θ = π/5

Substituting n = 2 and k = 0, we get the third solution:

θ = 2π/5

Substituting n = 3 and k = 0, we get the fourth solution:

θ = 3π/5

Substituting n = 4 and k = 0, we get the fifth solution:

θ = 4π/5

Substituting n = 5 and k = 0, we get the sixth solution:

θ = π

Therefore, the multiple-angle equation has 6 solutions, which correspond to the angles:

0, π/5, 2π/5, 3π/5, 4π/5, π

Converting these to degrees, we get:

0°, 36°, 72°, 108°, 144°, 180°

The answer is the choice E, which contains these angles in radians as:

11eacce6_1ebf_4fe8_9f49_8904c2094ebd_TB7896_11

Solve the Multiple-Angle Equation. ​

Solve the Multiple-Angle Equation.