Compute the measure of the angle between 0 and 360 degrees s

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. FInd the degrees

A: y=0.7 in Quadrant II

B:y= -0.9 in Quadrant III.

C: y=-0.1 in Quadrant IV.

Solution

Here we have that as it is a unit circle, its radius is 1.

Now in first part A, when terminal side intersects the circle at a point (x,0.7), then it forms a right triangle there with 90 degree angle at y axis having its base as distanc e0.7 on y axis and its hypotenues as 1, so on applying the formula that

cos A = base/hypotenues,          where A is base angle, we get

cos A = 0.7/1 = 0.7

or A = arccos(0.7) = 45.6

and thus this terminal side makes an angle of 90+45.6 = 135.6 degree with positive x axis.

Answer of A) part.

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Similarly when terminal side is in III Quad then, vertical distance or height of thus formed right triangle is 0.9

and thus on applying the formula that sin A = height/hypotenues, we have

sin A = 0.9/1 = 0.9

or A = arcsin(0.9) =64 degree

so the terminal side forms a total of 180+64 = 224 degree angle with positive x axis.

This is the answer of part B)

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AGain in third part C)

the height of thus formed right triangle is 0.1

so using same sine formula

sin A = 0.1/1 = 0.1

or A= arcsin(0.1) =5.74

Now as any angle is IV quad is given as (2pi-A )

so its terminal side makes an angle of 360 - 5.74 = 354.26 degree with positive x axis.

Answer of part C)