The point is on the terminal side of an angle in standard position. Determine the exact value of .

Question

The point  is on the terminal side of an angle in standard position. Determine the exact value of  .

Quizplus · Accepted Answer

The point given in the question is on the terminal side of an angle in standard position, which means the angle is formed by the initial side (positive x-axis) and the terminal side (the given point). We need to find the exact value of this angle, which is the angle between the initial side and the terminal side measured in the counterclockwise direction. 
To find the exact value of the angle, we need to use trigonometric functions. Let's label the coordinates of the given point as (x,y). Then, we have:

cos(theta) = x/r, sin(theta) = y/r, where r is the distance from the origin to the point (x,y).

We can find r using the Pythagorean theorem:

r^2 = x^2 + y^2

r = sqrt(x^2 + y^2)

Now, we can substitute into the equations for cos(theta) and sin(theta):

cos(theta) = x/sqrt(x^2 + y^2)
sin(theta) = y/sqrt(x^2 + y^2)

To find the exact value of the angle, we need to use the inverse trigonometric function. Since cos(theta) is positive and sin(theta) is negative (because the point is in the fourth quadrant where x is positive and y is negative), we can use the formula:

theta = 2pi - arctan(y/x)

Substituting in the values of x and y, we get:

theta = 2pi - arctan(-239/1211)

Using a calculator, we can simplify this to:

theta = 11eb80b1_aff6_1a05_a291_a7c89a40480e_TB8633_11

Therefore, the best choice is (D).

The Point Is on the Terminal Side of an Angle