DWG NO. 4.1 — Lesson 1 of 6

Angles, Degree and Radian Measure

Unit 4: Trigonometric Functions and the Unit Circle · ~20 min

Objective Sketch an angle in standard position, identify coterminal angles, and convert between degree and radian measure.

Every angle in this unit is drawn in standard position: its vertex sits at the origin, and its initial side lies along the positive x-axis. The terminal side is wherever the angle ends up after rotating — counterclockwise for a positive angle, clockwise for a negative one. That single convention is what lets an angle be paired with a specific point on a circle, which is exactly what the rest of this unit depends on.

Coterminal angles

Two angles are coterminal if they share the same terminal side — they end up in the same place, just by rotating a different amount. Since one full trip around a circle is 360° (or 2π radians), you can always find a coterminal angle by adding or subtracting a multiple of a full rotation.

In degrees: θ and θ ± 360°n are coterminal for any integer n.
In radians: θ and θ ± 2πn are coterminal for any integer n.

Degrees and radians

A radian is defined directly from the circle itself: it's the angle created when the arc length swept out equals the radius. Walk exactly one radius-length along the circle's edge, and the angle from center to start-point to end-point measures 1 radian — roughly 57.3°. Because the circumference of a circle is 2πr, a full rotation measures 2π radians, giving the conversion between the two systems.

Degrees to radians: multiply by \(\frac{\pi}{180}\)
Radians to degrees: multiply by \(\frac{180}{\pi}\)

One radian: the angle formed when the arc swept out has the same length as the radius r

Common conversions

Degrees	0°	30°	45°	60°	90°	180°	270°	360°
Radians	0	\(\frac{\pi}{6}\)	\(\frac{\pi}{4}\)	\(\frac{\pi}{3}\)	\(\frac{\pi}{2}\)	π	\(\frac{3\pi}{2}\)	2π

Worked Example 1 · Degrees to radians

ProblemConvert 150° to radians.

1Multiply by \(\frac{\pi}{180}\): 150 · \(\frac{\pi}{180}\) = \(\frac{150\pi}{180}\).

2Reduce the fraction: \(\frac{150}{180}\) = \(\frac{5}{6}\), so the result is \(\frac{5\pi}{6}\).

150° = \(\frac{5\pi}{6}\) radians

Worked Example 2 · Radians to degrees

ProblemConvert \(\frac{7\pi}{4}\) radians to degrees.

1Multiply by \(\frac{180}{\pi}\): \(\frac{7\pi}{4}\) · \(\frac{180}{\pi}\) = \(\frac{7\cdot180}{4}\).

27 · 180 = 1260, and 1260 ÷ 4 = 315.

\(\frac{7\pi}{4}\) radians = 315°

Worked Example 3 · Coterminal angles

ProblemFind a positive and a negative angle coterminal with 400°.

1400° is already more than one rotation. Subtract 360° once: 400° − 360° = 40°. That's the positive coterminal angle in [0°, 360°).

2For a negative one, subtract 360° again: 40° − 360° = −320°.

400° is coterminal with 40° and with −320°

Guided practice

1.Convert 210° to radians.
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210 · \(\frac{\pi}{180}\) = \(\frac{210\pi}{180}\) = \(\frac{7\pi}{6}\).
2.Convert \(\frac{5\pi}{3}\) radians to degrees.
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\(\frac{5\pi}{3}\) · \(\frac{180}{\pi}\) = \(\frac{5\cdot180}{3}\) = \(\frac{900}{3}\) = 300°.
3.Find a positive coterminal angle for −70° in the interval [0°, 360°).
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−70° + 360° = 290°.
4.Find a coterminal angle for \(\frac{\pi}{6}\) by adding 2π.
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\(\frac{\pi}{6}\) + 2π = \(\frac{\pi}{6}\) + \(\frac{12\pi}{6}\) = \(\frac{13\pi}{6}\).
5.Explain why a radian is a "unitless" way to measure angle, in terms of its definition.
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A radian is defined as a ratio of arc length to radius — both are lengths, so their units cancel. That's why radian measure is just a plain number (like 1, or π), unlike degrees, which is an arbitrary division of a circle into 360 parts.

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