Question

Compute the measure of the angle between 0 and 360 degrees swept counterclockwise from 3 o'clock position on the unit circle whose terminal ray intersects the circle at the point with given y-coordinate and in the given quadrant.x=0.5 in Quadrant I.θ=  degrees   x=-0.8 in Quadrant II.θ= degrees   x=-0.1 in Quadrant III.θ=  degrees   x=0.8 in Quadrant IV.θ=  degrees

Answer 1

ANSWERS

x=0.5 in Quadrant I.

θ = 60 degrees



x=-0.8 in Quadrant II.

θ = 143.13 degrees



x=-0.1 in Quadrant III.

θ = 264.26 degrees



x=0.8 in Quadrant IV.

θ = 323.13 degrees



Step-by-step explanation

For all the angles we know the length of the adjacent side of the triangle and the length of the hypotenuse - which is the radius of the circle.

For the first triangle in red, the angle is:

`\begin{gathered} \cos \theta=(0.5)/(1) \\ \cos \theta=0.5 \\ \theta=\cos ^(-1)0.5 \\ \theta=60º \end{gathered}`

For the second triangle, in green, we can find the supplementary angle of θ:

`\begin{gathered} \cos (180º-\theta)=(0.8)/(1) \\ 180º-\theta=\cos ^(-1)0.8 \\ 180º-\theta=36.87º \\ \theta=180º-36.87º \\ \theta=143.13º \end{gathered}`

For the third triangle, in light-blue, the angle we'll find is (θ - 180º):

`\begin{gathered} \cos (\theta-180º)=(0.1)/(1) \\ \theta-180º=\cos ^(-1)0.1 \\ \theta=180º+84.26º \\ \theta=264.26º \end{gathered}`

And for the last triangle, in pink, the angle we'll find from the triangle is (360º-θ):

`\begin{gathered} \cos (360º-\theta)=(0.8)/(1) \\ 360º-\theta=\cos ^(-1)0.8 \\ \theta=360º-36.87º \\ \theta=323.13º \end{gathered}`