I have to retype this.
For which angles theta,
with 0 degrees less than/equal to theta less than/equal to 180 degrees, is sin 3theta = (equal) -1 (negative 1)?
a) 30 and 150 degrees
b) 60 and 120 degrees
c) 90 degrees
d) 0 and 120 degrees
How do I set this up?
To find the values of theta for which sin 3theta equals -1, we need to set up an equation and solve for theta.
First, let's use the identity sin(3theta) = 3sin(theta) - 4sin^3(theta). Setting this equal to -1 gives us:
3sin(theta) - 4sin^3(theta) = -1
To simplify this equation, let's substitute y for sin(theta):
3y - 4y^3 = -1
Now, we have a cubic equation in terms of y. To solve for y, we'll rearrange the equation to the form of 4y^3 - 3y + 1 = 0.
Next, we can factor or use numerical methods to solve this cubic equation. However, since the answer options only provide discrete values for theta, we can solve the equation by checking each possible value.
Let's go through each option and plug it into the equation sin 3theta = -1 (or equivalently, 3y - 4y^3 = -1).
a) For theta = 30 degrees:
We have y = sin(30 degrees) = 1/2
Substituting this into the equation: 3(1/2) - 4(1/2)^3 = 3/2 - 4/8 = 3/2 - 1/2 = 1
Since 1 is not equal to -1, option a) is incorrect.
b) For theta = 60 degrees:
We have y = sin(60 degrees) = √3/2
Substituting this into the equation: 3(√3/2) - 4(√3/2)^3 = 3√3/2 - 4(3√3/8) = 3√3/2 - 3√3/2 = 0
Since 0 is not equal to -1, option b) is incorrect.
c) For theta = 90 degrees:
We have y = sin(90 degrees) = 1
Substituting this into the equation: 3(1) - 4(1)^3 = 3 - 4 = -1
Since -1 is equal to -1, option c) is a possible solution.
d) For theta = 120 degrees:
We have y = sin(120 degrees) = √3/2
Substituting this into the equation: 3(√3/2) - 4(√3/2)^3 = 3√3/2 - 4(3√3/8) = 3√3/2 - 3√3/2 = 0
Since 0 is not equal to -1, option d) is incorrect.
Therefore, the correct answer is option c) 90 degrees.