Angles and the unit circle

Quadrants, coterminal angles, degrees and radians, arcs on the unit circle. 5 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution. In all answers \(k, m \in \mathbb{Z}\).

Problem 1 Quadrants

In which quadrant does each angle lie?

a) \(160^\circ\) b) \(250^\circ\) c) \(310^\circ\) d) \(830^\circ\) e) \(-470^\circ\) f) \(\frac{11\pi}{4}\) g) \(-\frac{13\pi}{6}\)

Show solution

First find the coterminal angle between \(0^\circ\) and \(360^\circ\) (or between \(0\) and \(2\pi\)) by adding or subtracting full turns.

a) \(90^\circ < 160^\circ < 180^\circ\): II.
b) \(180^\circ < 250^\circ < 270^\circ\): III.
c) \(270^\circ < 310^\circ < 360^\circ\): IV.
d) \(830^\circ - 2\cdot 360^\circ = 110^\circ\): II.
e) \(-470^\circ + 2\cdot 360^\circ = 250^\circ\): III.
f) \(\frac{11\pi}{4} = 2\pi + \frac{3\pi}{4}\), and \(\frac{\pi}{2} < \frac{3\pi}{4} < \pi\): II.
g) \(-\frac{13\pi}{6} + 4\pi = \frac{11\pi}{6}\), and \(\frac{3\pi}{2} < \frac{11\pi}{6} < 2\pi\): IV.

Problem 2 Primary directed angle

Find the primary directed angle (the coterminal angle in \([0^\circ, 360^\circ)\), or in \([0, 2\pi)\)) of:

a) \(1000^\circ\) b) \(-800^\circ\) c) \(-1500^\circ\) d) \(\frac{47\pi}{6}\) e) \(-\frac{29\pi}{4}\)

Show solution

Use \(\beta = \alpha + k\cdot 360^\circ\) (or \(\beta = \alpha + 2k\pi\)) and choose \(k\) so that the result is in the interval.

a) \(1000^\circ - 2\cdot 360^\circ = 280^\circ\).
b) \(-800^\circ + 3\cdot 360^\circ = 280^\circ\).
c) \(-1500^\circ + 5\cdot 360^\circ = 300^\circ\).
d) \(\frac{47\pi}{6} - 3\cdot 2\pi = \frac{47\pi - 36\pi}{6} = \frac{11\pi}{6}\).
e) \(-\frac{29\pi}{4} + 4\cdot 2\pi = \frac{-29\pi + 32\pi}{4} = \frac{3\pi}{4}\).

Problem 3 Degrees and radians

Use \(\dfrac{D}{180^\circ} = \dfrac{R}{\pi}\) to convert.

To radians: a) \(36^\circ\) b) \(150^\circ\) c) \(225^\circ\) d) \(-72^\circ\) e) \(20^\circ\)

To degrees: f) \(\frac{7\pi}{9}\) g) \(\frac{11\pi}{12}\) h) \(3\) (radians)

Show solution

Degrees to radians: \(R = \dfrac{D\cdot\pi}{180^\circ}\).

a) \(\frac{36\pi}{180} = \frac{\pi}{5}\); b) \(\frac{150\pi}{180} = \frac{5\pi}{6}\); c) \(\frac{225\pi}{180} = \frac{5\pi}{4}\); d) \(-\frac{72\pi}{180} = -\frac{2\pi}{5}\); e) \(\frac{20\pi}{180} = \frac{\pi}{9}\).

Radians to degrees: \(D = \dfrac{R\cdot 180^\circ}{\pi}\).

f) \(\frac{7}{9}\cdot 180^\circ = 140^\circ\); g) \(\frac{11}{12}\cdot 180^\circ = 165^\circ\); h) \(\frac{3\cdot 180^\circ}{\pi} \approx 171.89^\circ\).

Problem 4 Unit circle and real numbers

On the unit circle, a central angle corresponds to a real number on the number line (the arc length). Find this number, correct to three decimal places:

a) \(40^\circ\) b) \(210^\circ\) c) \(-50^\circ\) d) \(\frac{3\pi}{5}\) e) \(\frac{17\pi}{4}\)

Show solution

On the unit circle (\(r = 1\)) the arc length is equal to the angle in radians, so we only convert to radians.

a) \(40^\circ = \frac{2\pi}{9} \approx 0.698\)
b) \(210^\circ = \frac{7\pi}{6} \approx 3.665\)
c) \(-50^\circ = -\frac{5\pi}{18} \approx -0.873\)
d) \(\frac{3\pi}{5} \approx 1.885\)
e) \(\frac{17\pi}{4} \approx 13.352\)

Problem 5 Coterminal angles

a) Which pairs are coterminal? (i) \(75^\circ\) and \(-285^\circ\); (ii) \(130^\circ\) and \(490^\circ\); (iii) \(40^\circ\) and \(-40^\circ\); (iv) \(\frac{\pi}{3}\) and \(\frac{7\pi}{3}\); (v) \(\frac{5\pi}{4}\) and \(-\frac{3\pi}{4}\); (vi) \(100^\circ\) and \(280^\circ\).

b) Write all angles coterminal with \(65^\circ\) and list those between \(-720^\circ\) and \(720^\circ\).

Show solution

a) Two angles are coterminal when their difference is a multiple of \(360^\circ\) (of \(2\pi\)).

(i) \(75^\circ - (-285^\circ) = 360^\circ\): yes.
(ii) \(490^\circ - 130^\circ = 360^\circ\): yes.
(iii) the difference is \(80^\circ\): no.
(iv) the difference is \(2\pi\): yes.
(v) the difference is \(2\pi\): yes.
(vi) the difference is \(180^\circ\): no.

b) \(\beta = 65^\circ + k\cdot 360^\circ\), \(k\in\mathbb{Z}\). Between \(-720^\circ\) and \(720^\circ\): \(-655^\circ,\ -295^\circ,\ 65^\circ,\ 425^\circ\).