Solve the equation 4sin^2 thiter +4cos thiter=5 for 0 degrees<=thiter<=360 degrees, give your answer in degrees
To solve the equation 4sin^2(x) + 4cos(x) = 5, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1. Rearranging this identity, we get sin^2(x) = 1 - cos^2(x).
Substituting this into the equation above, we have:
4(1 - cos^2(x)) + 4cos(x) = 5
4 - 4cos^2(x) + 4cos(x) - 5 = 0
-4cos^2(x) + 4cos(x) - 1 = 0
Now, let's solve this quadratic equation for cos(x). We can use the quadratic formula:
cos(x) = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -4, b = 4, and c = -1. Substituting these values into the quadratic formula:
cos(x) = (-4 ± √(4^2 - 4(-4)(-1))) / (2(-4))
cos(x) = (-4 ± √(16 - 16)) / (-8)
cos(x) = (-4 ± √0) / (-8)
cos(x) = (-4 ± 0) / (-8)
Since the square root of 0 is 0, we have:
cos(x) = -4 / -8 = 1/2
Now, using the unit circle or a calculator, we can find the angles where cos(x) = 1/2. These angles are 60 degrees and 300 degrees (or -60 degrees).
Therefore, the solutions for x (in degrees) are x = 60° and x = 300° (or -60°).
To solve the equation 4sin^2 θ + 4cos θ = 5, we can use the trigonometric identity: sin^2 θ + cos^2 θ = 1.
Rearranging the equation, we have:
4sin^2 θ + 4(1 - sin^2 θ) = 5
Expanding and simplifying, we get:
4sin^2 θ + 4 - 4sin^2 θ = 5
Combining like terms:
4 - 5 = 4sin^2 θ - 4sin^2 θ
Simplifying further:
-1 = 0
Since the equation -1 = 0 is not true, there are no solutions for the given equation 4sin^2 θ + 4cos θ = 5.