Question

For the rotation 65 9 ∘ 659 ∘ , find the coterminal angle from 0 ∘ ≤ � < 36 0 ∘ 0 ∘ ≤θ<360 ∘ , the quadrant, and the reference angle. the coterminal angle is °, which lies in quadrant , with a reference angle of °.

Answer 1

425°

Explanation:

To find the coterminal angle for
`(65^\circ)`, We can add or subtract multiples of (360°) until we get an angle between (0°) and (360°).

`\implies \: \: (65^\circ + 360^\circ = 425^\circ)`

So, one coterminal angle is (425°).

To determine the quadrant, We can look at the original angle (65°):

- Quadrant I:
`\sf{(0^\circ < \theta < 90^\circ)}`

- Quadrant II:
`\sf(90^\circ < \theta < 180^\circ)`

- Quadrant III:
`\sf{(180^\circ < \theta < 270^\circ)}`

- Quadrant IV:
`\sf{(270^\circ < \theta < 360^\circ)}`

In this case, (65°) lies in Quadrant I.

Now, to find the reference angle, We can use the formula:

`\small{ \longrightarrow \: (\text{Reference Angle} = |\text{Angle} - 90^\circ * \text{Quadrant Number}|)}`

`\small{ \longrightarrow \: (\text{Reference Angle} = |65^\circ - 90^\circ * 1| = |65^\circ - 90^\circ| = |25^\circ| = 25^\circ)}`

So, the coterminal angle is (425°), which lies in Quadrant I, with a reference angle of (25°).