Degrees and Radians

📐 Degrees and Radians

Part 1 of 7 — Two Ways to Measure an Angle

Topics in This Part

🔑 Key Concept: A degree and a radian both answer the same question — "how much have we rotated?" — but they use different rulers. Degrees split a full turn into $360$ equal pieces; radians measure rotation by the arc length swept out on a circle of radius $1$.

What an Angle Measure Counts

Picture a ray pinned at the center of a circle, sweeping counterclockwise. The angle records how far it has turned — not how long the ray is, not how big the circle is, just the amount of rotation.

Because rotation has no built-in unit, humans invented two:

Both are valid. A right angle is $90^\circ$ and $\dfrac{\pi}{2}$ radians — same rotation, two labels.

💡 Think of it like distance: $1$ mile and $1.609$ km describe the same gap. Degrees and radians describe the same angle.

The Degree System

The degree is the older, more familiar unit. A full circle is divided into $360$ degrees, written $360^\circ$.

Why $360$? The number is convenient — it's divisible by $2, 3, 4, 5, 6, 8, 9, 10, 12, \ldots$ — so many common angles come out whole:

💡 Degrees are great for everyday talk ("turn $90$ degrees"), but they are arbitrary — there's no deep reason a circle has $360$ pieces rather than $400$. Radians fix that.

Concept Check 🎯

The Radian System

A radian is defined by the circle itself, not by a chosen number. Take a circle of radius $r$ and walk along its edge a distance of exactly $r$. The angle you swept out is one radian.

$1 \text{ radian} = \text{the angle whose arc length equals the radius}$

How many radians fit in a full circle? The full circumference is $2\pi r$, and each radian "uses up" an arc of length $r$, so a full turn is

$\frac{2\pi r}{r} = 2\pi \text{ radians} \approx 6.283 \text{ radians.}$

That gives the master link between the two systems:

$\boxed{\,2\pi \text{ rad} = 360^\circ \quad\Longleftrightarrow\quad \pi \text{ rad} = 180^\circ\,}$

🔑 Memorize this: $\pi$ radians $= 180^\circ$. Every conversion in this lesson comes from that single equation.

Concept Check 🎯

Why Radians Exist

If degrees already work, why bother? Because radians are the natural unit for circles, and they make later mathematics far cleaner:

• Arc length is just $s = r\theta$ — no messy constants — only when $\theta$ is in radians.
• Calculus of $\sin$ and $\cos$ behaves nicely (e.g. $\frac{d}{dx}\sin x = \cos x$) only in radians.
• The unit circle, which you'll use all through trigonometry and calculus, is built on radians.

⚠️ Heads up: From here on, when an angle is written with no degree symbol — like $\frac{\pi}{3}$ or $2$ — it is in radians. A bare number means radians; only the little $^\circ$ means degrees.

Next up: converting fluently between the two.

Fill In the Basics 🧮

Enter just the number (no symbols).

1) A full turn is how many degrees? $\,?$ 2) A full turn is how many radians, written as a multiple of $\pi$ (form like 2pi)? $\,?$ 3) A straight line (half turn) is how many degrees? $\,?$

📐 Degrees and Radians

Part 2 of 7 — Converting Between Units

🔑 The Master Equation: $\pi \text{ rad} = 180^\circ$. To convert, multiply by a fraction equal to $1$ — either $\dfrac{\pi}{180^\circ}$ or $\dfrac{180^\circ}{\pi}$ — choosing the one that cancels the unit you have.

Degrees → Radians

To go from degrees to radians, multiply by $\dfrac{\pi}{180^\circ}$ (degrees on the bottom cancel the degrees you start with):

$\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180^\circ}$

Worked Example: convert $60^\circ$

$60^\circ \cdot \frac{\pi}{180^\circ} = \frac{60\pi}{180} = \frac{\pi}{3}$

Worked Example: convert $270^\circ$

$270^\circ \cdot \frac{\pi}{180^\circ} = \frac{270\pi}{180} = \frac{3\pi}{2}$

💡 Shortcut: divide the degree measure by $180$, reduce the fraction, and slap a $\pi$ on top. $60/180 = 1/3 \Rightarrow \frac{\pi}{3}$.

Radians → Degrees

Going the other way, multiply by $\dfrac{180^\circ}{\pi}$ (this time $\pi$ cancels):

$\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180^\circ}{\pi}$

Worked Example: convert $\dfrac{\pi}{4}$

$\frac{\pi}{4} \cdot \frac{180^\circ}{\pi} = \frac{180^\circ}{4} = 45^\circ$

Worked Example: convert $\dfrac{5\pi}{6}$

$\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi} = \frac{5 \cdot 180^\circ}{6} = \frac{900^\circ}{6} = 150^\circ$

💡 The $\pi$'s cancel, leaving plain numbers. Whenever you see $\pi$ vanish, you know you're heading toward degrees.

Concept Check 🎯

Degrees → Radians 🧮

Convert each to radians. Write your answer as a fraction times $\pi$ using the form like 5pi/6 (a number, then pi, then /, then a number). If it reduces to a whole multiple of $\pi$, just write it (e.g. 2pi).

1) $30^\circ = \,?$ 2) $135^\circ = \,?$ 3) $360^\circ = \,?$

Radians → Degrees 🧮

Convert each to degrees. Enter just the number (no degree symbol).

1) $\dfrac{\pi}{6} = \,?$ 2) $\dfrac{3\pi}{2} = \,?$ 3) $\pi = \,?$

📐 Degrees and Radians

Part 3 of 7 — The Special Angles

🔑 Why this matters: A handful of angles appear constantly in trig and calculus. Knowing their degree and radian forms by heart — without converting each time — is the single biggest time-saver in the whole subject.

The Angles Worth Memorizing

These are the "first quadrant" benchmarks plus the axis angles. Learn the row that goes with each fraction of $\pi$:

💡 Pattern: the denominators $6, 4, 3$ go with $30^\circ, 45^\circ, 60^\circ$ — bigger angle, smaller denominator. $\frac{\pi}{6}$ is small, $\frac{\pi}{3}$ is twice as big.

Building the Rest from Multiples

Every other "nice" angle is a multiple of $30^\circ$ ($\frac{\pi}{6}$) or $45^\circ$ ($\frac{\pi}{4}$). Just count:

$\frac{\pi}{6},\ \frac{2\pi}{6}=\frac{\pi}{3},\ \frac{3\pi}{6}=\frac{\pi}{2},\ \frac{4\pi}{6}=\frac{2\pi}{3},\ \frac{5\pi}{6},\ \frac{6\pi}{6}=\pi$

In degrees that's $30, 60, 90, 120, 150, 180$ — steps of $30^\circ$.

Likewise the multiples of $45^\circ$ are $\frac{\pi}{4}, \frac{2\pi}{4}=\frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \ldots$ — degrees $45, 90, 135, 180, 225, \ldots$

⚠️ Always reduce. $\frac{2\pi}{6}$ should be written $\frac{\pi}{3}$, and $\frac{3\pi}{6}$ is $\frac{\pi}{2}$. Un-reduced fractions still convert correctly but look unfinished.

Match the Pairs 🔽

Pick the radian measure that equals each degree measure.

Concept Check 🎯

Recall the Special Angles 🧮

Fill in the matching measure.

1) $\dfrac{\pi}{3}$ in degrees $= \,?$ 2) $150^\circ$ in radians (form like 5pi/6) $= \,?$ 3) $\dfrac{7\pi}{6}$ in degrees $= \,?$

📐 Degrees and Radians

Part 4 of 7 — Coterminal Angles & Standard Position

🔑 Big idea: Many different angle measures point the terminal ray to the same place. Adding or subtracting a full turn — $360^\circ$ or $2\pi$ — lands you back where you started. Those are coterminal angles.

Standard Position

An angle is in standard position when:

• its vertex sits at the origin, and
• its initial side lies along the positive $x$-axis.

The terminal side is where the ray ends up after rotating.

• Counterclockwise rotation → positive angle.
• Clockwise rotation → negative angle.

So $+90^\circ$ points straight up, while $-90^\circ$ (or $-\frac{\pi}{2}$) points straight down.

💡 A negative angle isn't "smaller than nothing" — it just means you turned the other direction.

Coterminal Angles

Two angles are coterminal if their terminal sides land in the exact same spot. You get a coterminal angle by adding or subtracting whole turns:

$\theta \pm 360^\circ \qquad\text{or}\qquad \theta \pm 2\pi$

Worked Example (degrees)

Find a positive angle coterminal with $-50^\circ$: $-50^\circ + 360^\circ = 310^\circ$

Worked Example (radians)

Find a positive angle coterminal with $-\dfrac{\pi}{3}$: $-\frac{\pi}{3} + 2\pi = -\frac{\pi}{3} + \frac{6\pi}{3} = \frac{5\pi}{3}$

Worked Example (subtracting)

$420^\circ$ is more than a full turn; subtract one: $420^\circ - 360^\circ = 60^\circ$

⚠️ When you add $2\pi$ to a fraction, first rewrite $2\pi$ with the same denominator: $2\pi = \frac{6\pi}{3}$ when working with thirds.

Concept Check 🎯

Find a Coterminal Partner 🔽

Choose the smallest positive angle coterminal with each given angle.

Coterminal Practice 🧮

Give the smallest positive coterminal angle.

1) $-100^\circ$ → (degrees) $\,?$ 2) $480^\circ$ → (degrees) $\,?$ 3) $-\dfrac{\pi}{6}$ → (radians, form like 11pi/6) $\,?$

📐 Degrees and Radians

Part 5 of 7 — Arc Length & Sector Area

🔑 The payoff of radians: the arc length swept by an angle is simply $s = r\theta$ — but only when $\theta$ is measured in radians. This is the formula that makes radians worth learning.

Arc Length

On a circle of radius $r$, a central angle $\theta$ (in radians) sweeps an arc of length

$\boxed{\,s = r\theta\,}$

This works because $\theta$ in radians literally counts "how many radius-lengths of arc," so the arc is $r$ times that count.

Worked Example

A circle has radius $r = 5$ cm. Find the arc length cut off by a central angle of $\theta = \dfrac{\pi}{3}$.

$s = r\theta = 5 \cdot \frac{\pi}{3} = \frac{5\pi}{3} \approx 5.24 \text{ cm}$

⚠️ Convert first! If the angle is given in degrees, change it to radians before using $s=r\theta$. Using degrees here gives a wrong answer.

When the Angle Is in Degrees

Suppose $r = 12$ in and $\theta = 90^\circ$. First convert:

$90^\circ = \frac{\pi}{2} \text{ rad}$

Then apply the formula:

$s = r\theta = 12 \cdot \frac{\pi}{2} = 6\pi \approx 18.85 \text{ in}$

💡 Sanity check: $90^\circ$ is a quarter turn, so the arc should be a quarter of the circumference. Quarter of $2\pi(12) = 24\pi$ is $6\pi$. ✓ Same answer.

Concept Check 🎯

Sector Area

A sector is the "pizza slice" between two radii. Its area, with $\theta$ in radians, is

$\boxed{\,A = \frac{1}{2} r^2 \theta\,}$

This is just the fraction $\dfrac{\theta}{2\pi}$ of the whole circle's area $\pi r^2$:

$A = \frac{\theta}{2\pi}\cdot \pi r^2 = \frac{1}{2}r^2\theta$

Worked Example

Radius $r = 6$, central angle $\theta = \dfrac{\pi}{4}$:

$A = \frac{1}{2}(6)^2\cdot\frac{\pi}{4} = \frac{1}{2}\cdot 36 \cdot \frac{\pi}{4} = \frac{36\pi}{8} = \frac{9\pi}{2} \approx 14.14$

💡 Notice both formulas need radians. That's the whole reason precalculus pivots to radians before calculus.

Pick the Right Result 🔽

Use $s = r\theta$ and $A = \frac{1}{2}r^2\theta$ with $\theta$ in radians.

Arc & Sector Drill 🧮

Use $s = r\theta$ and $A = \frac{1}{2}r^2\theta$ with $\theta$ in radians. Give exact answers as a multiple of $\pi$ (form like 6pi or 9pi/2).

1) $r = 3$, $\theta = \dfrac{2\pi}{3}$. Arc length $s = \,?$ 2) $r = 10$, $\theta = \dfrac{\pi}{5}$. Arc length $s = \,?$ 3) $r = 4$, $\theta = \dfrac{\pi}{2}$. Sector area $A = \,?$

📐 Degrees and Radians

Part 6 of 7 — Angular Speed & Real-World Rotation

🔑 Where this shows up: wheels, gears, turntables, ceiling fans, and orbits all rotate. Angular speed measures how fast an angle changes; linear speed measures how fast a point on the rim moves. Radians tie them together.

Angular Speed

Angular speed $\omega$ (omega) is the angle swept per unit time:

$\omega = \frac{\theta}{t} \quad\text{(radians per unit time)}$

Useful conversions

• $1$ revolution $= 2\pi$ radians $= 360^\circ$.
• "Revolutions per minute" (RPM) becomes radians per minute by multiplying by $2\pi$.

Worked Example

A wheel spins at $3$ revolutions per second. Its angular speed is

$\omega = 3 \cdot 2\pi = 6\pi \text{ rad/s} \approx 18.85 \text{ rad/s}$

💡 Each revolution is one full $2\pi$ "lap" of angle, so multiply revolutions by $2\pi$ to get radians.

Linear Speed from Angular Speed

A point on the rim, distance $r$ from the center, moves at linear speed

$\boxed{\,v = r\omega\,}$

This is the time-rate version of $s = r\theta$: divide $s = r\theta$ by time $t$ to get $\dfrac{s}{t} = r\dfrac{\theta}{t}$, i.e. $v = r\omega$ (with $\omega$ in radians per unit time).

Worked Example

A bicycle wheel of radius $r = 0.35$ m turns at $\omega = 10$ rad/s. The bike's speed is

$v = r\omega = 0.35 \cdot 10 = 3.5 \text{ m/s}$

⚠️ $v = r\omega$ requires $\omega$ in radians per unit time. If you're given RPM, convert to rad/s first.

Concept Check 🎯

Set Up the Rotation Problem 🔽

A fan blade tip is $r = 0.5$ m from the center and the fan spins at $2$ rev/s.

Rotation Drill 🧮

Give exact answers as a multiple of $\pi$ (form like 6pi) unless told otherwise.

1) A gear turns at $4$ rev/s. Angular speed in rad/s $= \,?$ 2) $r = 2$ m, $\omega = 3$ rad/s. Linear speed $v = r\omega = \,?$ (plain number, m/s) 3) A wheel spins at $\frac{1}{2}$ rev/s. Angular speed in rad/s $= \,?$

📐 Degrees and Radians

Part 7 of 7 — Mixed Mastery & Exit Quiz

You can now (1) explain both unit systems, (2) convert fluently, (3) recall the special angles, (4) find coterminal angles, (5) compute arc length and sector area, and (6) handle angular and linear speed. Let's lock it in.

Quick Reference

⚠️ The #1 trap: using degrees in $s=r\theta$, $A=\frac{1}{2}r^2\theta$, or $v=r\omega$. Those three demand radians every time.

Mixed Practice 🎯

Final Setup Check 🔽

A disc of radius $r = 0.2$ m spins at $5$ revolutions per second.

Last Drill Before the Quiz 🧮

1) Convert $240^\circ$ to radians (form like 4pi/3) $= \,?$ 2) Smallest positive angle coterminal with $-\dfrac{\pi}{4}$ (form like 7pi/4) $= \,?$ 3) $r = 5$, $\theta = \dfrac{\pi}{2}$. Sector area $A = \frac{1}{2}r^2\theta = \,?$ (form like 25pi/4)

Exit Quiz ✅

Answer all three to finish the lesson.