1. Determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle.
theta = 4
I know how to find the terminal side when theta has pi in it (for example: 4pi/3), but I don't understand what just "4" means. 4 pi?
2. Find an angle between 0 and 2pi that is coterminal with the given angle.
11pi/7
How do I find an angle coterminal to 11pi/7 when that angle itself is already between 0 and 2pi?
1. When the angle is given as just "4", it is implied that the angle is measured in radians. In this case, "4" means 4 radians. Keep in mind that angles are often given in terms of π (pi) in trigonometry.
To determine the quadrant in which the terminal side of the angle is found, you can use the following guidelines:
- If the angle is between 0 and π/2 (0 and 90 degrees), it lies in the first quadrant.
- If the angle is between π/2 and π (90 and 180 degrees), it lies in the second quadrant.
- If the angle is between π and 3π/2 (180 and 270 degrees), it lies in the third quadrant.
- If the angle is between 3π/2 and 2π (270 and 360 degrees), it lies in the fourth quadrant.
Since we have a single value for theta (4), we need to convert it to degrees to determine the quadrant. To convert from radians to degrees, multiply the radian value by 180/π.
4 * (180/π) ≈ 229.18 degrees
Since 229.18 degrees lies between π and 3π/2 (180 and 270 degrees), the terminal side of the angle lies in the third quadrant.
To find the corresponding reference angle, subtract the angle from π (180 degrees) or 2π (360 degrees), depending on which quadrant the angle is in. In this case, the angle is in the third quadrant, so we subtract it from π (180 degrees).
Reference angle = π - 4 ≈ 3.14 - 4 ≈ -0.86 radians (approximately)
2. To find an angle coterminal with 11π/7 that is between 0 and 2π, you can add or subtract any integer multiple of 2π to the given angle. In other words, you can add or subtract 2π (360 degrees) to find other angles with the same terminal side.
To make sure the angle is between 0 and 2π, you can reduce or simplify the given angle. In this case, 11π/7 cannot be simplified further.
To find the coterminal angle between 0 and 2π, we can add or subtract multiples of 2π:
11π/7 + 2π ≈ (11π + 14π)/7 ≈ 25π/7
The angle 25π/7 is coterminal with 11π/7 and lies within 0 and 2π.