HL · Further trig 1 · Radian measure

Module 22 of 47Lesson 1 of 9 · HL · Further trig 1 · Radian measure

Radians, further trigonometry & matrix transformations · Essential understanding 1

Radian measure

IB AI AHL 3.7with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: A radian is another unit for angles. Instead of cutting a full turn into 360 pieces, we measure an angle by the length of arc it cuts off on a circle of radius 1. A full turn is

2\pi

radians, so

\pi

radians = 180°.

By the end of this lesson you can:

explain what one radian is
convert between degrees and radians, with exact answers in terms of $\pi$
remember the radian values of the common angles
switch the TI-Nspire CX II between degree and radian mode and avoid the classic mode mistake
use sine, cosine and tangent with angles given in radians

🧑🏾 Imagine this…

You walk round a circular pond in Maun. If the pond has radius 10 m, then 10 m of walking along the edge takes you through 1 radian of turn around the middle. Walk 20 m and you have turned 2 radians. Walk all the way round,

2\pi \times 10 \approx 62.8

m, and you have turned

2\pi

radians: one full turn. Radians simply count how many radius-lengths you have walked.

One radian: the arc is exactly as long as the radius.

Radian (rad)

the size of the angle at the centre of a circle when the arc is the same length as the radius.

Degree

one part of 360 equal parts of a full turn. It is a human invention: 360 divides nicely, but it has nothing to do with the circle itself.

Exact form

an angle written with

\pi

, such as

\dfrac{3\pi}{4}

. It is exact; the decimal 2.36 is only an approximation.

Radian mode

the calculator setting that makes sin, cos and tan expect radians. The word RAD appears at the top of the screen.

★ Converting (the only fact you need is 180° = π rad)

$360^\circ = 2\pi$ radians, so $180^\circ = \pi$ radians.
Degrees to radians: multiply by $\dfrac{\pi}{180}$ .
Radians to degrees: multiply by $\dfrac{180}{\pi}$ .
1 radian $= \dfrac{180}{\pi} \approx 57.3^\circ$ .
If the question says “exact”, leave $\pi$ in your answer. If it says “3 s.f.”, give a decimal.

The angles you meet most often. Learn the first quarter (0,

\dfrac{\pi}{6}

\dfrac{\pi}{4}

\dfrac{\pi}{3}

\dfrac{\pi}{2}

) and the rest follow by adding on

\dfrac{\pi}{2}

each time.

Common angles

Degrees	30°	45°	60°	90°	120°	135°	150°	180°	270°	360°
Radians	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$	$\dfrac{2\pi}{3}$	$\dfrac{3\pi}{4}$	$\dfrac{5\pi}{6}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$

⚠️ The mode mistake that costs the most marks

Your calculator does not know which unit you mean. If it is in degree mode and you type sin(1.2), it treats 1.2 as 1.2 degrees. If it is in radian mode and you type sin(30), it treats 30 as 30 radians and gives −0.988 instead of 0.5. Before every trig question, look at the top of the screen: RAD or DEG? In this unit you will mostly want RAD.

⚠️ Do not put a degree sign on a radian

Radians have no special symbol. Write

\dfrac{\pi}{3}

or 1.05, never

\dfrac{\pi}{3}^\circ

. If you write “rad” or nothing, both are fine.

💡 Quick sense-check

One radian is a bit less than 60° (it is 57.3°). So 2 radians is about 115°, and 3 radians is just under 180°, since

\pi \approx 3.14

. If your answer for 1.2 rad is 4000° something has gone wrong.

🧪 Try it yourselfRadians versus degrees

The line shows an angle in degrees (up) for each angle in radians (across). Drag the marker to 1.2: the height is about 68.8°, the answer to Example 2(c). Notice that the gradient is always 57.3: every radian is worth 57.3°. Now drag to π ≈ 3.14 and see 180°.

Worked examples

Example 1.1Degrees to radians, exact

Convert to radians, giving exact answers in terms of

\pi

: (a) 60° (b) 135° (c) 210° (d) 15°. Then give 75° as a decimal to 3 s.f.

🧠 Think it through

Multiply each angle by $\dfrac{\pi}{180}$ and simplify the fraction.
A faster way: 180° is π, so 60° is a third of π, 15° is a twelfth of π.
For the decimal, type the multiplication on the GDC (it is in radian mode).

✍️ By hand

(a)

60 \times \dfrac{\pi}{180} = \dfrac{\pi}{3}

(b)

135 \times \dfrac{\pi}{180} = \dfrac{135\pi}{180} = \dfrac{3\pi}{4}

(c)

210 \times \dfrac{\pi}{180} = \dfrac{7\pi}{6}

(d)

15 \times \dfrac{\pi}{180} = \dfrac{\pi}{12}

🔢 On your TI-Nspire CX II

Make sure the top of the screen says RAD. Type 75*π/180 and press enter ↵💻 Enter. (The π symbol is on the π💻 type pi key.)
The GDC gives only decimals: 75° = 1.31 rad

✅ Answer (a)

\dfrac{\pi}{3}

(b)

\dfrac{3\pi}{4}

(c)

\dfrac{7\pi}{6}

(d)

\dfrac{\pi}{12}

. 75° =

\dfrac{5\pi}{12}

= 1.31 radians (3 s.f.).

Example 1.2Radians to degrees

Convert to degrees: (a)

\dfrac{2\pi}{5}

rad (b)

\dfrac{5\pi}{6}

rad (c) 1.2 rad (d) 0.5 rad. Give (c) and (d) to 3 s.f.

🧠 Think it through

Multiply by $\dfrac{180}{\pi}$ . When π is in the angle, it cancels exactly.
For a decimal angle, type it into the calculator.

✍️ By hand

(a)

\dfrac{2\pi}{5} \times \dfrac{180}{\pi} = \dfrac{360}{5} = 72^\circ

(b)

\dfrac{5\pi}{6} \times \dfrac{180}{\pi} = 150^\circ

(c)

1.2 \times \dfrac{180}{\pi} = 68.8^\circ

🔢 On your TI-Nspire CX II

Type 1.2*180/π and 0.5*180/π.
68.7549 → 68.8°; 28.6479 → 28.6°

✅ Answer (a) 72° (b) 150° (c) 68.8° (d) 28.6°

Check with the sense-check tip: 1.2 rad should be a little more than 60°. It is 68.8°. ✓

Example 1.3A turning clock hand

The hour hand of a clock moves from 3:00 to 5:30. Through what angle does it turn? Give the answer in degrees and as an exact number of radians.

🧠 Think it through

The hour hand goes right round (360°) in 12 hours, so it turns $\dfrac{360}{12} = 30^\circ$ every hour.
From 3:00 to 5:30 is 2.5 hours.
Convert to radians: 30° = $\dfrac{\pi}{6}$ , so multiply $\dfrac{\pi}{6}$ by 2.5.

✍️ By hand

Angle

= 2.5 \times 30^\circ = 75^\circ

In radians:

2.5 \times \dfrac{\pi}{6} = \dfrac{5\pi}{12}

🔢 On your TI-Nspire CX II

Check: type 5*π/12 enter ↵💻 Enter → 1.309. That is the same as 75* $\dfrac{\pi}{180}$ from Example 1. ✓

✅ Answer 75°, which is

\dfrac{5\pi}{12}

radians (1.31 rad).

Notice how the radian answer is built from simple steps: a whole turn is 2π, so 1 hour is

\dfrac{2\pi}{12} = \dfrac{\pi}{6}

Example 1.4Trig functions in radian mode

Use your GDC to find: (a) sin(0.7) (b)

\sin\left(\dfrac{2\pi}{5}\right)

\sin\left(\dfrac{\pi}{6}\right)

in the wrong mode, and what the right answer is.

🧠 Think it through

Set RAD mode first, then type each one.
For (d), you know sin 30° = 0.5, and $\dfrac{\pi}{6}$ radians is 30°.

🔢 On your TI-Nspire CX II

Check that the top of the screen says RAD. If it says DEG, change it: home💻 click ⌂ on keypad ▸ Settings ▸ Document Settings, then set Angle to Radian.
Type sin(0.7), sin(2*π/5), cos(1.9) and sin(π/6).
📍 Where to find it on the TI-Nspire
⌨️ sin, cos, tan: click trig on the keypad, or type sin(
Angle must be Radian
RAD mode: $\sin\left(\dfrac{\pi}{6}\right)$ = 0.5 exactly as expected
DEG mode: the calculator thinks $\dfrac{\pi}{6}$ ≈ 0.52 degrees

✅ Answer (a) 0.644 (b) 0.951 (c) −0.323 (d) In RAD mode

\sin\left(\dfrac{\pi}{6}\right)

= 0.5 ✓. In DEG mode the GDC would give 0.00914, which is wrong for this question.

The GDC is only as clever as its mode. Make “check RAD” the first thing you do on every trig question.

Example 1.5A ladder against a wall

A ladder of length 12 m leans against a vertical wall in Gaborone. The angle between the ladder and the ground is 0.9 radians. Find (a) how high up the wall the ladder reaches (b) how far the foot of the ladder is from the wall. Give both to 3 s.f.

🧠 Think it through

Draw the right-angled triangle: hypotenuse 12 m, angle 0.9 rad at the ground.
Height is the side opposite the angle → sine: height = 12 sin(0.9).
Distance along the ground is adjacent → cosine: 12 cos(0.9).
Radian mode! 0.9 is a radian value.

✍️ By hand

(a)

\sin 0.9 = \dfrac{\text{height}}{12}

, so height

= 12\sin 0.9

(b)

\cos 0.9 = \dfrac{\text{distance}}{12}

, so distance

= 12\cos 0.9

🔢 On your TI-Nspire CX II

In RAD mode type 12*sin(0.9) and 12*cos(0.9).
📍 Where to find it on the TI-Nspire
⌨️ sin, cos, tan: click trig on the keypad, or type sin(
Check with Pythagoras: it gives 12 ✓

✅ Answer (a) 9.40 m (b) 7.46 m

0.9 rad is about 51.6°, so the ladder is quite steep. A sensible-looking answer: height bigger than the distance from the wall.

Challenge questions

⭐ Challenge 1.6Convert and comparePaper 1 style

(a) Write 112.5° in radians as an exact fraction of π. (b) Write

\dfrac{7\pi}{4}

radians in degrees. (c) Which is larger, 1.9 radians or 110°?

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Multiply by $\dfrac{\pi}{180}$ for (a); by $\dfrac{180}{\pi}$ for (b).
To compare in (c), put both in the same unit.

🔢 On your TI-Nspire CX II

Type 1.9*180/π enter ↵💻 Enter → 108.857. Compare with 110.

112.5° =

\dfrac{5\pi}{8}

= 1.96 rad

✅ Answer (a)

112.5 \times \dfrac{\pi}{180} = \dfrac{5\pi}{8}

(b) 315° (c) 1.9 rad = 109° (3 s.f.), so 110° is larger.

⭐ Challenge 1.7The sprinkler armPaper 2 style

A garden sprinkler arm in Kasane swings through an angle of 2.1 radians, then back again. It makes 3 swings in a row, all in the same direction, without stopping.

(a) Convert 2.1 radians to degrees (3 s.f.).
(b) What fraction of a full turn is one swing? Give a decimal to 3 s.f.
(c) Find the total angle turned after the 3 swings, in radians.
(d) How many complete turns is this, and what angle is left over?

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

(a) Multiply by $\dfrac{180}{\pi}$ .
(b) A full turn is 2π radians: $\text{fraction} = \dfrac{2.1}{2\pi}$ .
(c) 3 × 2.1.
(d) Subtract 2π (one full turn) from the total. Is what is left bigger than 2π again?

✍️ By hand

(b)

\dfrac{2.1}{2\pi} = 0.334

(d)

6.3 - 2\pi = 0.0168

rad

🔢 On your TI-Nspire CX II

Type 2.1*180/π, 2.1/(2*π), 6.3-2*π.

✅ Answer (a) 120° (b) 0.334 of a turn (c) 6.3 rad (d) One complete turn with 0.0168 radians (about 0.96°) left over.

Next: Lesson 2 · Arc length & sector area in radians

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · Further trig 1 · Radian measure

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Radian measure

Common angles

Worked examples

Challenge questions