07.5 - Radians

For θ = 3π/5, which of the following correctly identifies the quadrant of the terminating ray and the reference angle?

• Quadrant II; Reference angle = 2π/5

• Quadrant I; Reference angle = 3π/5

• Quadrant III; Reference angle = π/5

• Quadrant IV; Reference angle = 3π/5

Answer explanation

For θ = 3π/5, the angle is in Quadrant II since it is between π/2 and π. The reference angle is found by subtracting θ from π: π - 3π/5 = 2π/5. Thus, the correct choice is Quadrant II; Reference angle = 2π/5.

For θ = 11π/9, which of the following correctly identifies the quadrant of the terminating ray and the reference angle?

• Quadrant III; Reference angle = 2π/9

• Quadrant II; Reference angle = π/9

• Quadrant IV; Reference angle = π/9

• Between Quadrant II and III; Reference angle = π/9

Answer explanation

For θ = 11π/9, it lies in Quadrant III since it is between π and 3π/2. The reference angle is found by subtracting π: 11π/9 - π = 2π/9. Thus, the correct choice is Quadrant III; Reference angle = 2π/9.

For θ = 5π/18, identify the quadrant in which the terminating ray lies and the reference angle.

• Quadrant II; Reference angle = 5π/18

• Quadrant I; Reference angle = 5π/18

• Quadrant III; Reference angle = 13π/18

• Quadrant IV; Reference angle = π/18

Answer explanation

The angle θ = 5π/18 is in the first quadrant since it is between 0 and π/2. The reference angle is the angle itself, which is 5π/18.

For θ = 8π/5, identify the quadrant in which the terminating ray lies and the reference angle in radians.

• Quadrant I, Reference angle: π/5

• Quadrant IV, Reference angle: 2π/5

• Quadrant II, Reference angle: 3π/5

• Quadrant III, Reference angle: 4π/5

Answer explanation

The angle θ = 8π/5 is in Quadrant IV since it is between 3π/2 and 2π. The reference angle is found by subtracting 2π from θ, giving us 8π/5 - 2π = 2π/5. Thus, the correct answer is Quadrant IV, Reference angle: 2π/5.

For θ = 3π/2, which of the following correctly identifies the quadrant or quadrants for the terminating ray, and the reference angle?

• Quadrantal; reference angle is π/2

• Quadrant II; reference angle is π/2

• Quadrant IV; reference angle is π/2

• Quadrant III; reference angle is π

Answer explanation

For θ = 3π/2, the angle lies on the negative y-axis, making it a quadrantal angle. The reference angle is the angle from the x-axis, which is π/2. Thus, the correct choice is 'Quadrantal; reference angle is π/2'.

For θ = 7π/3, identify the quadrant in which the terminating ray lies and the reference angle in radians.

• First quadrant, reference angle π/3

• Third quadrant, reference angle π/4

• Fourth quadrant, reference angle π/6

• Second quadrant, reference angle 2π/3

Answer explanation

For θ = 7π/3, we can subtract 2π (or 6π/3) to find the equivalent angle in the first quadrant, which is π/3. Thus, the angle lies in the first quadrant with a reference angle of π/3.

For θ = 3π/4, which of the following correctly identifies the quadrant of the terminating ray and the reference angle?

• Quadrant II; Reference angle = π/4

• Quadrant III; Reference angle = π/4

• Quadrant II; Reference angle = π/2

• Quadrant I; Reference angle = 3π/4

Answer explanation

For θ = 3π/4, the angle is in Quadrant II. The reference angle is found by subtracting θ from π, giving π - 3π/4 = π/4. Thus, the correct choice is Quadrant II; Reference angle = π/4.

For θ = 15π/8, identify the quadrant in which the terminating ray lies and the reference angle in radians.

• Fourth quadrant, reference angle = π/8

• Third quadrant, reference angle = 7π/8

• First quadrant, reference angle = π/4

• Second quadrant, reference angle = 3π/8

Answer explanation

For θ = 15π/8, subtract 2π (or 16π/8) to find the equivalent angle in the fourth quadrant: 15π/8 - 16π/8 = -π/8. The reference angle is the positive angle, which is π/8. Thus, the answer is fourth quadrant, reference angle = π/8.

For θ = 17π/5, identify the quadrant in which the terminating ray lies and the reference angle in radians.

• Second quadrant, reference angle = 3π/5

• Third quadrant, reference angle = 2π/5

• First quadrant, reference angle = π/5

• Fourth quadrant, reference angle = 4π/5

Answer explanation

To find the quadrant for θ = 17π/5, reduce it by subtracting 2π (or 10π/5), resulting in 7π/5, which lies in the third quadrant. The reference angle is 7π/5 - π = 2π/5. Thus, the correct answer is third quadrant, reference angle = 2π/5.

For θ = 5π/9, which quadrant does the terminating ray lie in, and what is the reference angle?

• Quadrant II; Reference angle = 5π/9

• Quadrant I; Reference angle = 5π/9

• Quadrant II; Reference angle = 4π/9

• Quadrant III; Reference angle = 4π/9

Answer explanation

The angle θ = 5π/9 is between π/2 and π, placing it in Quadrant II. The reference angle is found by subtracting θ from π: π - 5π/9 = 4π/9. Thus, the correct choice is Quadrant II; Reference angle = 4π/9.