OMPT drill — Trigonometry

Angles and radians

Radians are the native language of the OMPT trig chapters. Converting fluently and knowing the unit-circle landmarks by heart is non-negotiable. Keep a unit circle sketch next to you for the first pass, then redo the set without it.

Tested inOMPT-B OMPT-D OMPT-G

Lesson

A radian measures an angle by arc length: one radian is the angle you get when you walk a distance of one radius along the rim of a circle. Since the full rim is $2 π$ radii long, a full turn is $2 π$ radians, a half turn is $π$ , a right angle is $\frac{π}{2}$ . Nothing mystical: it is just a more natural unit than the degree, which owes its 360 to Babylonian astronomy rather than to any property of circles.

The OMPT trig chapters speak radians almost exclusively, so fluency is the entry fee here. And I do mean fluency: seeing $\frac{5 π}{6}$ and knowing it lives just short of the negative $x$ -axis, without converting to 150° first. Students who translate everything back into degrees carry a permanent speed tax through three whole topics. The conversion factor is a single fact, $180° = π$ radians, and everything else in this lesson is practice at using it until you no longer need to.

Problem 1Numeric answer

A circular sector has radius $9$ and area $\frac{27 π}{4}$ . Find the central angle $θ$ in radians. Round to two decimals.

Show the worked solution

Answer: 0.52

I start from the sector area formula $A = \frac{1}{2} r^{2} θ$ . Plugging in gives $\frac{27 π}{4} = \frac{1}{2} \cdot 81 \cdot θ$ , so $θ = \frac{27 π}{4} \cdot \frac{2}{81} = \frac{π}{6} \approx 0.52$ rad.

Problem 2Multiple choice

Which angle is coterminal with $- \frac{5 π}{3}$ ?

A $\frac{π}{3}$
B $\frac{2 π}{3}$
C $\frac{5 π}{3}$
D $- \frac{π}{3}$

Show the worked solution

Answer: A — $\frac{π}{3}$

Coterminal angles differ by a full turn of $2 π$ . I add one turn: $- \frac{5 π}{3} + 2 π = \frac{π}{3}$ . That lands in the first quadrant, which matches the terminal side of the original angle.

Problem 3Numeric answer

A bicycle wheel of radius $0.35$ m spins at $240$ revolutions per minute. How fast, in m/s, does a point on the rim move? Round to two decimals.

Show the worked solution

Answer: 8.8

First I convert to angular speed: $240$ rev/min is $240 \cdot 2 π$ radians per minute, so $ω = \frac{480 π}{60} = 8 π$ rad/s. Linear speed is $v = r ω = 0.35 \cdot 8 π = 2.8 π \approx 8.80$ m/s.

Problem 4Multiple choice

What is $\frac{11 π}{12}$ in degrees?

A $15 5^{\circ}$
B $16 5^{\circ}$
C $172. 5^{\circ}$
D $19 5^{\circ}$

Show the worked solution

Answer: B — $16 5^{\circ}$

I multiply by $\frac{18 0 ^{\circ}}{π}$ : $\frac{11 π}{12} \cdot \frac{18 0 ^{\circ}}{π} = 11 \cdot 1 5^{\circ} = 16 5^{\circ}$ . Thinking of $\frac{π}{12}$ as $1 5^{\circ}$ makes twelfths quick to convert.

Problem 5Spot the error

A student converts $10 0^{\circ}$ to radians. Step 1: radians and degrees are proportional, with $18 0^{\circ} = π$ rad. Step 2: so $10 0^{\circ} = 100 \cdot \frac{180}{π}$ . Step 3: this gives approximately $5729.6$ rad. Which step contains the error?

Show the worked solution

Answer: Step 2

Step 1 states the correct proportion. The slip is in Step 2: to go from degrees to radians you multiply by $\frac{π}{180}$ , not $\frac{180}{π}$ . The correct value is $100 \cdot \frac{π}{180} = \frac{5 π}{9} \approx 1.75$ rad. A sanity check helps here: $10 0^{\circ}$ is just over a quarter turn, so the answer must be a bit under $2$ .

Problem 6Numeric answer

A pendulum of length $1.2$ m swings through an angle of $4 0^{\circ}$ . What arc length, in metres, does the tip trace in one swing? Round to two decimals.

Show the worked solution

Answer: 0.84

Arc length needs the angle in radians: $4 0^{\circ} = \frac{40 π}{180} = \frac{2 π}{9}$ . Then $s = r θ = 1.2 \cdot \frac{2 π}{9} = \frac{2.4 π}{9} \approx 0.84$ m.