Convert $60^\circ$ to radians and $\frac{5\pi}{6}$ to degrees.

$60^\circ = \frac{\pi}{3}$ and $\frac{5\pi}{6} = 150^\circ$

Convert $135^\circ$ and $-270^\circ$ to radians, and $\frac{7\pi}{4}$ to degrees.

$\frac{3\pi}{4}$, $-\frac{3\pi}{2}$ and $315^\circ$

Convert $2$ radians to degrees and $50^\circ$ to radians, each to two decimal places.

$2$ rad $\approx 114.59^\circ$ and $50^\circ = \frac{5\pi}{18} \approx 0.87$ rad

A bicycle wheel of radius $35$ cm turns at $150$ revolutions per minute. Find its angular speed in radians per second and the bike's speed in km/h, each to one decimal place.

$\omega = 5\pi \approx 15.7$ rad/s; the bike moves at about $19.8$ km/h

Grade 12 · MHF4U · Advanced Functions

MHF4U: Radian Measure

A radian is the angle you get when the arc of a circle is exactly as long as its radius. It is the natural unit for angles, and every trig function in this course is written in radians from here on.

How It Works

Going once around a circle is an arc of length $2\pi r$ , which is $2\pi$ radii, so a full turn is $2\pi$ radians. That one fact gives the conversion: $2\pi$ rad $= 360^\circ$ , so $\pi$ rad $= 180^\circ$ .

Degrees to radians: multiply by $\frac{\pi}{180}$ . Radians to degrees: multiply by $\frac{180}{\pi}$ . Keep $\pi$ in the answer whenever the degrees are a nice fraction of $180$ ; a decimal is only for angles like $2$ rad or $50^\circ$ that do not reduce.

The ones to know by heart, since they come up in every unit after this one: $30^\circ = \frac{\pi}{6}$ , $45^\circ = \frac{\pi}{4}$ , $60^\circ = \frac{\pi}{3}$ , $90^\circ = \frac{\pi}{2}$ , $180^\circ = \pi$ , $270^\circ = \frac{3\pi}{2}$ . Multiples follow: $\frac{5\pi}{6}$ is five lots of $30^\circ$ , so $150^\circ$ .

An angle with no unit written is in radians. $\sin 2$ means the sine of $2$ radians, about $115^\circ$ , not $2^\circ$ . Set your calculator to radian mode before anything in this course.

Arc length and speed. Because a radian is defined by arc over radius, an angle $\theta$ in radians cuts an arc of length $a = r\theta$ . The formula only works in radians. The same idea gives angular speed $\omega$ in radians per second, and linear speed $v = r\omega$ at the rim of anything that spins.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Convert $60^\circ$ to radians and $\frac{5\pi}{6}$ to degrees.
Show solution
1. $60 \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3}$ .
2. $\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150$ .
Answer: $60^\circ = \frac{\pi}{3}$ and $\frac{5\pi}{6} = 150^\circ$
Easy
Convert $135^\circ$ and $-270^\circ$ to radians, and $\frac{7\pi}{4}$ to degrees.
Show solution
1. $135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}$ , dividing top and bottom by $45$ .
2. $-270 \times \frac{\pi}{180} = -\frac{3\pi}{2}$ .
3. $\frac{7\pi}{4} \times \frac{180}{\pi} = 7 \times 45 = 315$ .
Answer: $\frac{3\pi}{4}$ , $-\frac{3\pi}{2}$ and $315^\circ$
Medium
Convert $2$ radians to degrees and $50^\circ$ to radians, each to two decimal places.
Show solution
1. $2 \times \frac{180}{\pi} = \frac{360}{\pi} \approx 114.59$ , so $2$ rad is about $114.59^\circ$ . Neither $2$ nor $\frac{360}{\pi}$ simplifies, so a decimal is the right form here.
2. $50 \times \frac{\pi}{180} = \frac{5\pi}{18} \approx 0.87$ rad.
Answer: $2$ rad $\approx 114.59^\circ$ and $50^\circ = \frac{5\pi}{18} \approx 0.87$ rad
Medium
A sector of a circle of radius $6$ cm has a central angle of $120^\circ$ . Find the arc length, exactly and to one decimal place.
Show solution
1. Convert first: $120^\circ = \frac{2\pi}{3}$ . The arc formula needs radians.
2. $a = r\theta = 6 \times \frac{2\pi}{3} = 4\pi$ .
3. $4\pi \approx 12.6$ cm. A check: $120^\circ$ is a third of the circle, and a third of the circumference $12\pi$ is $4\pi$ .
Answer: $4\pi \approx 12.6$ cm
Test level
A bicycle wheel of radius $35$ cm turns at $150$ revolutions per minute. Find its angular speed in radians per second and the bike's speed in km/h, each to one decimal place.
Show solution
1. One revolution is $2\pi$ rad, so $150$ rev/min is $150 \times 2\pi = 300\pi$ rad/min.
2. Per second: $\omega = \frac{300\pi}{60} = 5\pi \approx 15.7$ rad/s.
3. The rim moves at $v = r\omega = 35 \times 5\pi = 175\pi$ cm/s $\approx 549.8$ cm/s, and the bike moves as fast as its rim.
4. Convert: $549.8$ cm/s $= 5.498$ m/s, and multiplying by $3.6$ gives $19.8$ km/h.
Answer: $\omega = 5\pi \approx 15.7$ rad/s; the bike moves at about $19.8$ km/h

Learn It on Martian Lab

Martian Lab teaches this in the Trigonometry lesson of its MHF4U Trigonometric Ratios unit. Watch it, then take the Trigonometric Ratios practice tests to see if it stuck.

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Free Trigonometric Ratios practice test, with solutions

More Trigonometric Ratios topics: Proving Trig Identities · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Solving Trig Equations · Laws of Logarithms · Solving Logarithmic Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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