Exercises — Radian measure — definition, conversion formula degrees ↔ radians

$180^{\circ }=\pi \text{rad}$ ::: master identity — the whole page hangs off this. deg → rad ::: multiply by $\frac{\pi }{180}$ (number shrinks). rad → deg ::: multiply by $\frac{180}{\pi }$ (number grows). arc length ::: $s=r\theta$ with $\theta$ in radians. sector area ::: $A=\frac{1}{2}{r}^2\theta$ with $\theta$ in radians.

We want radians, so put $\pi$ on top: multiply by $\frac{\pi }{180}$. $45\times \frac{\pi }{180}=\frac{45\pi }{180}=\frac{\pi }{4}\text{rad}$ Sanity: the number shrank from $45$ to $\approx 0.785$ — correct direction for deg → rad. ✓

We want degrees, so put $180$ on top: multiply by $\frac{180}{\pi }$. $\frac{\pi }{6}\times \frac{180}{\pi }=\frac{180}{6}=30^{\circ }$ The $\pi$ upstairs cancels the $\pi$ downstairs — a tell you did it right. ✓

Full turn: the arc is the whole circumference $s=2\pi r$, so $\theta =s/r=2\pi r/r=2\pi \text{rad}$. Half turn is exactly half of that: $\pi \text{rad}$ (which is the master identity $180^{\circ }=\pi$). ✓

Step 1 — convert, because $s=r\theta$ demands radians. $\theta =30\times \frac{\pi }{180}=\frac{\pi }{6}\text{rad}$ Step 2 — apply $s=r\theta$ (the definition $\theta =s/r$ rearranged). $s=6\times \frac{\pi }{6}=\pi \approx 3.14\text{cm}$ Look at the figure: the arc (pale-yellow) is one $\frac{\pi }{6}$-slice of the rim. ✓

Use $A=\frac{1}{2}{r}^2\theta$ with $\theta =\frac{\pi }{6}$ (already converted above). $A=\frac{1}{2}\times 6^2\times \frac{\pi }{6}=\frac{1}{2}\times 36\times \frac{\pi }{6}=3\pi \approx 9.42{\text{cm}}^2$ Why this formula: the sector is a fraction $\frac{\theta }{2\pi }$ of the whole disc $\pi r^2$; the algebra collapses to $\frac{1}{2}{r}^2\theta$. ✓

Go straight to the definition $\theta =s/r$ — no conversion needed, we're asked for radians. $\theta =\frac{s}{r}=\frac{10}{4}=2.5\text{rad}$ A half turn is $\pi \approx 3.14\text{rad}$, so $2.5\text{rad}$ is smaller than a half turn. ✓

Convert (rad → deg, so $\times \frac{180}{\pi }$): $2\times \frac{180}{\pi }=\frac{360}{\pi }\approx 114.59^{\circ }$ Since $114.59^{\circ }>100^{\circ }$, $2\text{rad}$ is larger. Analysis note: one radian is $\approx 57.3^{\circ }$, so two of them land near $114.6^{\circ }$ — a fast mental estimate that confirms the exact value. ✓

Convert: $\frac{7\pi }{6}\times \frac{180}{\pi }=\frac{7\times 180}{6}=210^{\circ }$ Quadrant analysis: Quadrant I is $0$–$90^{\circ }$, II is $90$–$180^{\circ }$, III is $180$–$270^{\circ }$, IV is $270$–$360^{\circ }$. Since $180^{\circ }<210^{\circ }<270^{\circ }$, the angle sits in Quadrant III. See Trigonometric Functions of Any Angle for why the quadrant fixes the signs of $\sin ,\cos$. ✓

One full revolution $=2\pi \text{rad}=360^{\circ }$. Multiply by $5$: $5\times 2\pi =10\pi \text{rad}\approx 31.42\text{rad},5\times 360=1800^{\circ }.$ Why radians don't "reset": unlike a clock face, we let the count keep growing — this is exactly the language of Circular Motion, where total swept angle matters. ✓

Step 1 — solve $A=\frac{1}{2}{r}^2\theta$ for $\theta$ (rearrange the sector formula): $\theta =\frac{2A}{r^2}=\frac{2\times 12}{4^2}=\frac{24}{16}=1.5\text{rad}$ Step 2 — convert to degrees ($\times \frac{180}{\pi }$): $1.5\times \frac{180}{\pi }=\frac{270}{\pi }\approx 85.94^{\circ }$ Step 3 — arc length via $s=r\theta$ (uses the radian value, not the degree value): $s=4\times 1.5=6\text{cm}$ The figure shows this fat sector: nearly a quarter-turn, arc $6\text{cm}$ long. ✓

Step 1 — convert the angle, since $s=r\theta$ needs radians: $\theta =40\times \frac{\pi }{180}=\frac{2\pi }{9}\text{rad}\approx 0.6981\text{rad}$ Step 2 — solve $s=r\theta$ for $r$: $r=\frac{s}{\theta }=\frac{50}{2\pi /9}=\frac{50\times 9}{2\pi }=\frac{450}{2\pi }=\frac{225}{\pi }\approx 71.62\text{m}$ Why divide: $s=r\theta$ says arc = radius $\times$ angle, so radius = arc $÷$ angle. ✓ See Arc Length and Sector Area.

Convert $\alpha$ to degrees (easier than converting $\beta$ to a fraction of $\pi$): $\alpha =\frac{5\pi }{12}\times \frac{180}{\pi }=\frac{5\times 180}{12}=\frac{900}{12}=75^{\circ }$ Now compare in one unit: $75^{\circ }$ vs $80^{\circ }$ ⟹ $\beta$ is larger. Lesson: never compare a raw radian number to a raw degree number — convert to a common unit first. ✓

Start from the two radian freebies: $s=r\theta$ and $A=\frac{1}{2}{r}^2\theta$. From the first, $\theta =\frac{s}{r}$. Substitute into the second: $A=\frac{1}{2}{r}^2\cdot \frac{s}{r}=\frac{1}{2}rs.$ So $A=\frac{1}{2}sr$ — the sector's area is "half base $\times$ height" with the arc playing the role of the base and the radius the height (a curved echo of a triangle's area). Verify with 4.1: there $s=6\text{cm}$, $r=4\text{cm}$, so $A=\frac{1}{2}\times 6\times 4=12{\text{cm}}^2$ — matching the given area. ✓

(a) In $4\text{s}$ it sweeps angle $\theta =3\text{rad/s}\times 4\text{s}=12\text{rad}$. Distance along the rim is arc length: $s=r\theta =2\times 12=24\text{m}$. (b) One revolution is $2\pi \text{rad}$, so revolutions $=\frac{12}{2\pi }=\frac{6}{\pi }\approx 1.91$ revolutions. Why radians make this painless: $s=r\theta$ has no conversion factor — that clean link is precisely why angular speed is quoted in rad/s in Circular Motion. ✓

Because $\theta =s/r$ makes an angle a pure number (length ÷ length, dimensionless), $\sin \theta$ takes a plain number in and gives a plain number out — no hidden scale. If instead you measured in degrees, ${\theta }_{\mathrm{deg}}=\frac{180}{\pi }{\theta }_{\text{rad}}$, so a degree is a stretched copy of the radian. Any derivative "with respect to degrees" then drags along a chain-rule factor $\frac{\pi }{180}$: $\frac{d}{d{\theta }_{\mathrm{deg}}}\sin {\theta }_{\mathrm{deg}}=\frac{\pi }{180}\cos {\theta }_{\mathrm{deg}}.$ Radians are the unique scale where that factor is exactly $1$, so the rule is clean. This underpins both Small Angle Approximation ($\sin \theta \approx \theta$) and Derivatives of Trig Functions. ✓