Evaluate where C is the right half of the circle

Question

Evaluate  where C is the right half of the circle

Quizplus · Accepted Answer

The given expression represents the tangent of an angle whose vertex lies on the circle and whose rays intersect the x-axis at the given point. Since C is the right half of the circle, the angle whose vertex lies on C and whose rays intersect the x-axis at the given point is in the second quadrant. Thus, we can use the reference angle in the first quadrant, which is complement to the angle in the second quadrant. The reference angle is the angle whose tangent is 11eaae29_f436_4231_87fe_51bd301065d6_TB2067_11_TB2234_11.

Let x be the reference angle in the first quadrant.

tan(x) = 11eaae29_f436_4231_87fe_51bd301065d6_TB2067_11_TB2234_11 
tan(x) = sin(x)/cos(x)

Let r be the radius of the circle.

sin(x) = sqrt(r^2 - (distance from point to x-axis)^2) 
cos(x) = distance from point to y-axis

The distance from the point to x-axis is r/2 and the distance from the point to y-axis is sqrt(3)r/2.

sin(x) = sqrt(r^2 - (r/2)^2) = sqrt(3)r/2 
cos(x) = sqrt(3)r/2

tan(x) = (sqrt(3)r/2)/(sqrt(3)r/2) = 1

Thus, the reference angle is 45 degrees or pi/4 radians.

Since the angle we want is in the second quadrant, its measure is pi - pi/4 = 3pi/4 radians.

tan(3pi/4) = -1, since the tangent is negative in the second quadrant.

Therefore, evaluating the expression gives:

11eaae29_f436_4230_87fe_a9563345008b_TB2067_11 = -1

The only choice that gives a hexadecimal value of -1 is C: 
11eaae29_f436_9054_87fe_b9c80fc811cf_TB2067_11

Evaluate Where C Is the Right Half of the Circle