Given the following point on the unit circle, find the angle, to the nearest tenth of a degree (if necessary), of the terminal side through that point, 0, degrees, is less than or equal to, theta, is less than, 360, degrees, .0
∘
≤θ<360
∘
.
P, equals, left bracket, minus, start fraction, square root of, 5, end square root, divided by, 4, end fraction, comma, minus, start fraction, square root of, 11, end square root, divided by, 4, end fraction, right bracket
P=(−
4
5
,−
4
11
)
To find the angle of the terminal side passing through the point P, we first need to find the reference angle.
The reference angle is the angle θ' between the terminal side and the x-axis. We can find it using the coordinates of point P:
x = -√5/4
y = -√11/4
We can then calculate the reference angle as follows:
tan(θ') = y/x
tan(θ') = (-√11/4) / (-√5/4)
tan(θ') = √11 / √5
tan(θ') = √(11/5)
tan(θ') = √(11/5)
So, our reference angle θ' is the arctan of √(11/5):
θ' = arctan(√(11/5))
θ' ≈ 58.5 degrees
Since point P is in the third quadrant, we need to subtract this angle from 180 degrees to find the angle in standard position:
θ = 180 - θ'
θ = 180 - 58.5
θ = 121.5 degrees
Therefore, the angle of the terminal side passing through point P is approximately 121.5 degrees.