Question

which of these would be coterminal with a rotation of \\(1000^\circ\\)?

i. 280 degrees
ii. \\(\frac{50}{9}\pi\\) radians
iii. \\(-1000\\) degrees

• i only
• i and ii only
• ii and iii only
• i, ii, and iii

Explanation:

Define coterminal angles

Coterminal angles share the same terminal side.
They differ by integer multiples of \(360^\circ\) or \(2\pi\) radians.
We express this as \(\theta_1 - \theta_2 = 360^\circ \cdot k\) for some integer \(k\).

Analyze Statement I

We check if \(280^\circ\) is coterminal with \(1000^\circ\).
Subtract the two angles:

$$1000^\circ - 280^\circ = 720^\circ$$

Since \(720^\circ = 2 \cdot 360^\circ\), they differ by an integer multiple of \(360^\circ\).
Thus, Statement I is coterminal.

Analyze Statement II

Convert \(\frac{50}{9}\pi\) radians to degrees to compare.
Multiply by the conversion factor \(\frac{180^\circ}{\pi}\):

$$\frac{50}{9}\pi \cdot \frac{180^\circ}{\pi} = 50 \cdot 20^\circ = 1000^\circ$$

Since \(\frac{50}{9}\pi\) radians is exactly equal to \(1000^\circ\), they share the same terminal side.
Thus, Statement II is coterminal.

Analyze Statement III

We check if \(-1000^\circ\) is coterminal with \(1000^\circ\).
Subtract the two angles:

$$1000^\circ - (-1000^\circ) = 2000^\circ$$

Divide by \(360^\circ\) to check for an integer multiple:

$$\frac{2000^\circ}{360^\circ} = \frac{50}{9} \approx 5.56$$

Since this is not an integer, Statement III is not coterminal.

Determine the correct option

Only Statements I and II are coterminal with \(1000^\circ\).
This matches the option "I and II only".

Answer:

• I only
• I and II only (Correct answer)
• II and III only
• I, II, and III