Question

For each angle below, determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle theta¯ .

θ= -1π/3 is found in quadrant ____ and theta¯ = _______
θ= -1π/4 is found in quadrant ____ and theta¯ = _______
θ= -5π/6 is found in quadrant ____ and theta¯ = _______
θ= 5 is found in quadrant ____ and theta¯ = _______

Answer 1

For each angle:

1.
`\( \theta = -(\pi)/(3) \)` is in Quadrant III with
`\( \theta_{\text{ref}} = (\pi)/(3) \).`

2.
`\( \theta = -(\pi)/(4) \)` is in Quadrant III with
`\( \theta_{\text{ref}} = (\pi)/(4) \)`.

3
`\( \theta = -(5\pi)/(6) \)`is in Quadrant III with
`\( \theta_{\text{ref}} = (\pi)/(6) \).`

4.
`\( \theta = 5 \)` is in Quadrant I with
`\( \theta_{\text{ref}} = 5 \).`

1.
`\( \theta = -(\pi)/(3) \)`

This angle is in Quadrant III (Negative x-axis, Negative y-axis).

The reference angle
`\( \theta_{\text{ref}} \)` is
`\( (\pi)/(3) \)`.

2.
`\( \theta = -(\pi)/(4) \)`

Also in Quadrant III (Negative x-axis, Negative y-axis).

The reference angle
`\( \theta_{\text{ref}} \) is \( (\pi)/(4) \).`

3.
`\( \theta = -(5\pi)/(6) \)`

Once more, in Quadrant III (Negative x-axis, Negative y-axis).

The reference angle
`\( \theta_{\text{ref}} \) is \( (\pi)/(6) \)`.

4.
`\( \theta = 5 \)`

This angle is in Quadrant I (Positive x-axis, Positive y-axis).

As it's already in the first quadrant, the reference angle
`\( \theta_{\text{ref}} \) remains \( 5 \).`

Hence:

1.
`\( \theta = -(\pi)/(3) \)` is found in Quadrant III with
`\( \theta_{\text{ref}} = (\pi)/(3) \)`.

2.
`\( \theta = -(\pi)/(4) \)` is found in Quadrant III with
`\( \theta_{\text{ref}} = (\pi)/(4) \)`.

3.
`\( \theta = -(5\pi)/(6) \)` is found in Quadrant III with
`\( \theta = -(5\pi)/(6) \)`

4.
`\( \theta = 5 \)` is found in Quadrant I with
`\( \theta_{\text{ref}} = 5 \).`