Reference Angles and Coterminals│Please Help!

Question

Find the largest negative and the smallest positive coterminal angle on the interval [-2pi, 2pi] for each given angles below.

A. 29pi/12

B. 750°

Answer 1

To determine the largest negative and smallest positive coterminal angles within the interval [-2pi, 2pi] for each given angle, we either subtract or add multiples of 2pi or 360° (if dealing with degrees) until the angles are within the specified range.

A. 29pi/12

Begin by converting (29pi/12) into a decimal to better understand its relation to 2pi.

(29pi)/12 ≈ (29 x 3.1416)/12 ≈7.601

1. Given (2pi = 6.283), we can subtract (2pi) from (29pi/12) to obtain a coterminal angle within the interval [-2pi, 2pi].

(29pi/12) - (2pi) = (29pi - 24pi)/12 = 5pi/12

1. The angle (5pi/12) is a positive coterminal angle within [-pi, pi], thus the smallest positive coterminal angle is: 5pi/12

To find the largest negative coterminal angle, subtract another (2pi) from (5pi)/12:

1. (5pi/12)-2pi = (5pi -24pi)/12 = (-19pi)/12
2. Therefore, the largest negative coterminal angle is: (-19pi)/12

B. 750°

1. To find the coterminal angles in degrees, subtract 360° from 750° to bring it within the desired range:

750° - 360°= 390°

Since 390° is still above 360°, subtract another 360°:

390° - 360° = 30°

Hence, the smallest positive coterminal angle is 30°

1. To find the largest negative coterminal angle, subtract 360° from 30°:

30° - 360° = -330°

1. Thus, the largest negative coterminal angle is: -330°

Summary:

1. A. ( 29pi)/12:
2. Largest negative coterminal angle: ( -19pi/12)
3. Smallest positive coterminal angle: (5pi/12)
4. B. 750°:
5. Largest negative coterminal angle: -330°
6. Smallest positive coterminal angle: 30°

Answer 2

One complete loop around the circle in the positive direction (counterclockwise) is 2π radians or 360°. Likewise one complete loop around the circle in the negative direction (clockwise) is -2π or -360°.

A. So, to go 29π/12 is more than one complete loop (24π/12 is 2π) by 5π/12. So going in the positive direction by 5π/12 will put us in the same location. So 5π/12 is the smallest positive coterminal angle.

To find the largest negative angle, consider the 5π/12 and subtract 2π from it. 5π/12 - 24π/12 = -19π/12. Therefore -19π/12 is the largest negative coterminal angle.

B. 750° - 360° - 360° = 30° so 30° (which is π/6 radians) is the smallest positive coterminal angle.

30° - 360° = -330° so -330° (which is -11π/6 radians) is the largest negative coterminal angle.