Solve the equation for the given domain.
4cos^2 θ = 1 for 0 to 360 degrees.
I get 60, 120, 240, and 300. Are these correct? If so, how would I represent them with some sort of expression?
Your 4 solutions are correct for the primary domain
suppose we add the next 4, and we have
60, 120, 240, 300, 420, 480, 600, 660
notice that the 1st, the 3rd, the 5th, and the 7th, are obtained by adding consecutive multiples of 180°to 60°
and the 2nd, 4th, 6th, and 8th, are obtained by adding multiples of 180 to 120°
so a general solution would be:
60° + 180k , 120° + 180k° , where k is an integer.
Thank you very much!
To solve the equation 4cos^2θ = 1 for the given domain of 0 to 360 degrees, we can start by isolating the cosine term.
First, divide both sides of the equation by 4:
cos^2θ = 1/4
Next, take the square root of both sides:
cosθ = ±√(1/4)
Simplifying the right side gives:
cosθ = ±1/2
Now, let's find the values of θ that satisfy this equation. The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle, so we are looking for angles where the adjacent side is half the length of the hypotenuse. These angles correspond to the special triangles in the unit circle.
Using the unit circle, we can determine that the angles satisfying cosθ = ±1/2 are 60, 120, 240, and 300 degrees in the given domain of 0 to 360 degrees.
If you need to represent these angles with an expression, you can use θ = 60n, where n is an integer that can take on the values 0, 1, 2, or 3. This expression generates all the solutions: 60(0) = 0 degrees, 60(1) = 60 degrees, 60(2) = 120 degrees, 60(3) = 180 degrees, 60(4) = 240 degrees, and 60(5) = 300 degrees. However, since the given domain is 0 to 360 degrees, we only consider the solutions within that range, which are 60, 120, 240, and 300 degrees.