Question

Given sin theta =-0.7071 find the values of theta where 0 degrees <=theta <360 degrees.

Answer 1

The values of theta where sin theta = -0.7071 and 0 degrees <= theta < 360 degrees are 225 degrees and 315 degrees, representing the third and fourth quadrants respectively on the unit circle where sine is negative.

Step-by-step explanation:

Given sin theta = -0.7071, we are looking to find the values of theta within the range 0 degrees ≤ theta < 360 degrees. Since the sine function is negative, we know theta must be in either the third or fourth quadrant of the unit circle where the sine values are negative.

To find the reference angle, we calculate the inverse sine (arcsin) of the absolute value of the sine given. Thus, the reference angle, which is always positive, is arcsin(0.7071) which approximately equals 45 degrees. Therefore, theta could be either 180 + 45 degrees or 360 - 45 degrees, which are 225 degrees and 315 degrees respectively. These angles satisfy the condition in the third and fourth quadrants where the sine function is negative.

Steps to Find Theta:

1. Find the reference angle: arcsin(0.7071) = 45 degrees.
2. Since sin theta is negative, place the reference angle in the third and fourth quadrants.
3. Calculate the angles in these quadrants using the reference angle: 225 degrees (third quadrant) and 315 degrees (fourth quadrant).