find the angle between 0 degree and 360 of sin(theta)=-1 over square root of 2
To find the angle between 0 degrees and 360 degrees where sin(theta) = -1/√2, we can first find the reference angle and then consider the quadrants where sin(theta) is negative.
Step 1: Finding the reference angle
The reference angle is the positive acute angle formed between the terminal side of an angle in standard position and the x-axis. Since sin(theta) is negative for this problem, the reference angle will be between 180 degrees and 270 degrees.
Using the unit circle or a calculator, we can find that sin(225 degrees) = -√2/2. Therefore, the reference angle is 45 degrees.
Step 2: Considering the quadrants
Since sin(theta) = -1/√2, we are looking for angles in the second and third quadrants where the y-coordinate is negative.
In the second quadrant, the angle is between 90 degrees and 180 degrees.
In the third quadrant, the angle is between 180 degrees and 270 degrees.
Step 3: Combining the reference angle and quadrants
Add the reference angle (45 degrees) to the starting point of each quadrant:
Second Quadrant: 180 degrees + 45 degrees = 225 degrees
Third Quadrant: 270 degrees + 45 degrees = 315 degrees
Therefore, the angles between 0 degrees and 360 degrees where sin(theta) = -1/√2 are 225 degrees and 315 degrees.