An angle with an initial ray pointing in the 3-o'clock direction measures theta radians (where 0 less than or equal to 0 less than or equal to 2pi) the circles radius is 3 units long and the terminal point is located at (-2.69, -1.33)
a. The terminal point it how many radius lengths to the right of the circles vertical diameter
h= ____ radius lengths
b. When we evaluate cos^-1(h) the value returned is _____ radians
c. therefore, theta =
a. To find the horizontal distance from the vertical diameter of the circle to the terminal point, we can use the x-coordinate of the terminal point. Since the center of the circle is at the origin (since the radius is measured from the center), the vertical diameter of the circle is the line x = 0. So we want to find how many radius lengths to the right of this line the terminal point is. We can do this by dividing the x-coordinate by the length of the radius:
h = (-2.69) / 3 ≈ -0.8967
b. We want to find cos^-1(h). Since h is negative, we know that the terminal point is to the left of the vertical diameter, and so the angle theta is in the second quadrant. In this quadrant, cos is negative. Therefore, we want to find the angle whose cosine is equal to the absolute value of h. Using a calculator:
cos^-1(h) ≈ cos^-1(0.8967) ≈ 0.4728 radians
c. Since theta measures the angle between the initial ray and the terminal ray, and the initial ray points in the 3-o'clock direction (which is the positive x-axis), we know that the terminal ray lies in the second quadrant. Therefore, we can use the value of cos^-1(h) that we just found to determine theta:
theta = pi - cos^-1(h)
≈ pi - 0.4728
≈ 2.668 radians