Chapter 1 of 8

All 8 chapters
  1. 1Radians, readunderstood
  2. 2The unit circle and any angle, readunderstood
  3. 3Trig graphs and their changes, readunderstood
  4. 4Solving trig equations, readunderstood
  5. 5Identities, readunderstood
  6. 6Reciprocal and inverse trig functions, readunderstood
  7. 7Compound and double angles, readunderstood
  8. 8Combining sin and cos, readunderstood

M5.3.1 Chapter 1

Radians

Is there a better way to measure an angle than in degrees?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Change between degrees and radians
  • Find arc length and sector area using radians

Degrees come from an old habit of cutting a circle into 360 parts. Nothing about a circle makes 360 special. A radian is a unit that comes from the circle itself, and it makes the formulas for circles and for trigonometry short. From here on, most of the mathematics you meet uses radians.

Take a circle with radius r. Cut a piece of string of length r and bend it along the arc of the circle. The angle at the centre that the string covers is one radian. An arc as long as the radius cuts off one radian.

A sector of a circle with its centre at the bottom. Both radii are marked with a tick, and the arc is marked with a tick. The radius is labelled r, the arc is labelled r and the angle at the centre is labelled 1 rad. The sector is hatched.1 radrr
The arc has the same length as the radius, so the angle at the centre is 1 radian. The ticks show the two equal lengths.

This works for a circle of any size. A bigger circle has a longer radius and a longer arc, in the same ratio. Two radii of arc cut off 2 radians, and three radii cut off 3 radians. So an angle in radians is the number of radii that fit along its arc.

Definition. An angle θ at the centre of a circle has size θ = s ÷ r in radians. Here s is the length of the arc it cuts off and r is the radius. It is a plain number, so it has no unit. We may write "rad" to make it clear.

Now the whole circle. Its circumference is 2πr, which is 2π radii. So the full turn is 2π radians. Half a turn is π radians. This is the link between the two units: 2π radians = 360°, so π radians = 180°.

A circle with a centre O and axes named x and y. The axis ends are marked 0, π/2, π and 3π/2. The point at 60 degrees on the circle is named π/3, and the point at 135 degrees is named 3π/4.0π/2π3π/2xyπ/33π/4O
A quarter turn is π/2, a half turn is π, three quarters is 3π/2 and a full turn is 2π. The point at 60° is at angle π/3, and the point at 135° is at angle 3π/4.

One radian is 180° ÷ π, which is about 57.3°. To change units, use the fact that π radians = 180°. To change degrees to radians, multiply by π/180. To change radians to degrees, multiply by 180/π.

A table with the heads degrees and radians. The rows read 30 and π/6, 45 and π/4, 60 and π/3, 90 and π/2, 120 and 2π/3, 135 and 3π/4, 150 and 5π/6, 180 and π, 270 and 3π/2, 360 and 2π.degreesradians30π/645π/460π/390π/21202π/31353π/41505π/6180π2703π/23602π
The angles you will use most. Learn them as fractions of π: 30° is one sixth of 180°, so it is π/6.

Worked example

Change 150° to radians. Then change 7π/4 radians to degrees.

  1. We use π radians = 180°. So 1° is π/180 radians.
  2. Multiply 150 by π/180. This gives 150π/180.
  3. Cancel by 30. 150 ÷ 30 = 5 and 180 ÷ 30 = 6.
  4. So 150° = 5π/6 radians.
  5. For the other way, 1 radian is 180/π degrees. Multiply 7π/4 by 180/π. The π cancels.
  6. We are left with 7 × 180 ÷ 4. Since 180 ÷ 4 = 45, this is 7 × 45 = 315.
  7. So 7π/4 radians = 315°.

Answer: 150° = 5π/6 radians and 7π/4 radians = 315°.

Radians are usually written with no unit. If an angle has a degree sign, it is in degrees. If it has no sign, such as θ = 2 or x = π/3, it is in radians. From now on, an angle with no sign is in radians.

Arc length. The angle θ is the fraction θ/2π of a full turn. So the arc is that fraction of the circumference 2πr. The 2π cancels, and we are left with a short result: s = rθ. This also follows straight from the definition θ = s/r.

Sector area. A sector is a slice of a circle between two radii. It is the fraction θ/2π of the whole circle, which has area πr². So A = (θ/2π) × πr² = ½r²θ. Because s = rθ, we can also write A = ½rs.

A sector of a circle with its centre at the bottom. The radius is labelled r, the arc is labelled s and the angle at the centre is labelled θ. The sector is hatched.θrs
Arc length s = rθ. Sector area A = ½r²θ. Perimeter of the sector = 2r + rθ, because it has two straight sides as well as the arc. These need θ in radians.

Compare with degrees. If θ is in degrees, the arc is (θ/360) × 2πr and the sector is (θ/360) × πr². The extra π/180 sits in every formula. Radians remove it. That is why radians are used in calculus and in physics.

Worked example

A sector has radius 8 cm and angle 0.75 radians. Find its arc length and its area.

  1. The angle is in radians, so s = rθ and A = ½r²θ both apply.
  2. Arc length: s = 8 × 0.75 = 6 cm.
  3. Area: A = ½ × 8² × 0.75. Since 8² = 64, ½ × 64 = 32.
  4. Then 32 × 0.75 = 24. The area is 24 cm².
  5. Check with A = ½rs: ½ × 8 × 6 = 24. This agrees.

Answer: Arc length 6 cm and area 24 cm².

A sector of a circle with its centre at the bottom. The radius is labelled 8 cm, the arc is labelled s and the angle at the centre is labelled θ. The sector is hatched.θ8 cms
The sector of the worked example: radius 8 cm and θ = 0.75 radians.

Worked example

A sector has radius 6 cm and arc length 9 cm. Find its angle in radians, its area and its perimeter.

  1. We know r and s, so use θ = s ÷ r.
  2. θ = 9 ÷ 6 = 1.5 radians.
  3. Area: A = ½rs, since s is given. A = ½ × 6 × 9 = 27 cm².
  4. Check with A = ½r²θ: ½ × 36 × 1.5 = 18 × 1.5 = 27. This agrees.
  5. Perimeter: two radii and the arc. P = 6 + 6 + 9 = 21 cm.

Answer: θ = 1.5 radians, area 27 cm² and perimeter 21 cm.

A sector of a circle with its centre at the bottom. The radius is labelled 6 cm, the arc is labelled 9 cm and the angle at the centre is labelled θ. The sector is hatched.θ6 cm9 cm
Radius 6 cm and arc 9 cm. The angle is 9 ÷ 6 = 1.5 radians, about 86°.

A segment is the part of a circle between a chord and the arc. Its area is the sector minus the triangle formed by the two radii and the chord. The triangle has area ½r² sin θ, from the rule area = ½ab sin C that you know. So the segment has area ½r²θ − ½r² sin θ = ½r²(θ − sin θ). Use radian mode for sin θ.

Worked example

A chord AB of a circle with radius 10 cm cuts an angle of 1.4 radians at the centre. Find the area of the segment between the chord and the minor arc.

  1. Segment = sector − triangle. The angle is in radians, so the sector is ½r²θ.
  2. Sector: ½ × 10² × 1.4 = 50 × 1.4 = 70 cm².
  3. Triangle: ½r² sin θ = 50 × sin 1.4. The calculator is in radian mode, and sin 1.4 = 0.98545.
  4. So the triangle is 50 × 0.98545 = 49.27 cm².
  5. Segment = 70 − 49.27 = 20.73 cm². To 3 significant figures this is 20.7 cm².
  6. Check: the formula ½r²(θ − sin θ) = 50 × (1.4 − 0.98545) = 50 × 0.41455 = 20.73. This agrees.

Answer: The segment has area 20.7 cm².

A circle segment with the chord drawn across it. The two radii are drawn to the centre. The radius is labelled 10 cm, the chord is labelled chord and the angle at the centre is labelled θ. The segment between the chord and the arc is hatched.θ10 cmchord
The hatched segment is the sector with the triangle taken away.

Worked example

A sector has radius 5 cm and perimeter 16 cm. Find its angle in radians and its area.

  1. The perimeter is two radii plus the arc: 16 = 5 + 5 + s.
  2. So the arc is s = 16 − 10 = 6 cm.
  3. The angle is θ = s ÷ r = 6 ÷ 5 = 1.2 radians.
  4. The area is A = ½rs = ½ × 5 × 6 = 15 cm².
  5. Check with A = ½r²θ: ½ × 25 × 1.2 = 12.5 × 1.2 = 15. This agrees.

Answer: θ = 1.2 radians and the area is 15 cm².

A sector of a circle with its centre at the bottom. The radius is labelled 5 cm, the arc is labelled s and the angle at the centre is labelled θ. The sector is hatched.θ5 cms
Radius 5 cm and perimeter 16 cm. Take away the two radii, and 6 cm of arc is left.

Many exam questions join a sector to a triangle. The tangent to a circle at A is perpendicular to the radius OA. So if the tangent meets OB, extended, at T, triangle OAT has a right angle at A. Then AT = r tan θ and OT = r ÷ cos θ, with the calculator in radian mode.

Worked example

Sector OAB has radius 6 cm and angle 0.8 radians. The tangent at A meets OB, extended, at T. Find the perimeter and the area of the region bounded by AT, TB and the arc AB.

  1. Triangle OAT has a right angle at A, with OA = 6 cm and angle AOT = 0.8 radians.
  2. AT = 6 tan 0.8 = 6 × 1.0296 = 6.178 cm.
  3. OT = 6 ÷ cos 0.8 = 6 ÷ 0.6967 = 8.612 cm. So TB = OT − OB = 8.612 − 6 = 2.612 cm.
  4. The arc AB = rθ = 6 × 0.8 = 4.8 cm.
  5. The perimeter is AT + TB + arc AB = 6.178 + 2.612 + 4.8 = 13.59 cm, which is 13.6 cm.
  6. The area is triangle OAT − sector OAB = ½ × 6 × 6.178 − ½ × 6² × 0.8.
  7. This is 18.53 − 14.4 = 4.13 cm².

Answer: Perimeter 13.6 cm and area 4.13 cm²

A sector with its centre O at the bottom and its arc from A to B. The tangent at A, marked with a right angle, runs to T on the line OB extended. The points are named O, A, B and T. The radius is labelled 6 cm and the angle at O is labelled θ. The region between AT, TB and the arc is hatched.θ6 cmOABT
The angle θ is 0.8 radians. The point B lies on OT, 6 cm from O. The hatched region is triangle OAT with the sector OAB taken away.

For a small angle θ in radians, sin θ is almost θ, tan θ is almost θ, and cos θ is almost 1 − θ²/2. We write sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. At θ = 0.1: sin 0.1 = 0.0998, tan 0.1 = 0.1003 and cos 0.1 = 0.9950, while 1 − 0.1²/2 = 0.995. These hold only in radians, and only for small angles.

See it three ways

With things

Wrap a piece of string, as long as the radius, round the edge of a round tin lid. Mark where it ends. The angle from the centre of the lid to the two ends of the string is one radian. Two lengths of string give two radians.

As a picture

A sector of a circle with its centre at the bottom. Both radii are marked with a tick, and the arc is marked with a tick. The radius is labelled r, the arc is labelled r and the angle at the centre is labelled 1 rad. The sector is hatched.1 radrr

In words

The size of an angle in radians is the length of its arc divided by the radius.

In symbols

θ = s ÷ r, so s = rθ. For a full turn, s = 2πr, so θ = 2π.

Try it

Measure a radian with string

You need: a round lid or plate, a piece of thread, a ruler, a pencil.

  1. Measure the radius of the lid, from its centre to its edge, and cut a thread that long.
  2. Lay the thread along the edge of the lid. Mark the two ends of the thread on the edge.
  3. Mark the middle of the lid. Draw a line from it to each mark.
  4. Look at the angle between the two lines. It is about 57°, a little less than a sixth of the circle.

What you will see: The angle is the same whatever the size of the lid. A little more than six pieces of thread, 6.28 of them, go round the whole edge.

If you cannot try it, this is what happens.
A table with the heads pieces of thread and angle. The rows read 1 and 1 rad, 2 and 2 rad, 3 and 3 rad, 6.28 and 2π rad. The last row is a full turn.pieces of threadangle11 rad22 rad33 rad6.282π rad

A wrong idea many people have

The arc is s = rθ, and I can put the angle in degrees.

Why it feels right: The angle is the same angle, whatever unit we use, so it seems any unit should work. A sector of 60° looks like it has θ = 60.

The test that breaks it: Take r = 3 cm and θ = 60°. The rule would give s = 3 × 60 = 180 cm, but the whole circle has a circumference of only 2π × 3 = 18.85 cm. An arc cannot be longer than the circle.

What is true: The formulas s = rθ and A = ½r²θ hold only for θ in radians, because a radian is defined by arc ÷ radius. Change degrees to radians first: 60° = π/3, and then s = 3 × π/3 = π = 3.14 cm.

A wrong idea many people have

The perimeter of a sector is rθ.

Why it feels right: The arc is the curved edge, and that edge feels like the whole outline of the sector.

The test that breaks it: Draw a sector with radius 6 cm and θ = 1.5. The arc rθ is 9 cm. But the outline also has the two straight sides, each 6 cm.

What is true: The perimeter is 2r + rθ. For this sector, 6 + 6 + 9 = 21 cm. Read each question to see whether it asks for the arc or for the perimeter.

A wrong idea many people have

The calculator mode does not matter for sin and cos, because a sine is just a number.

Why it feels right: The buttons are the same in every mode, and sin 30 gives the same kind of answer.

The test that breaks it: Type sin 1. In degree mode it shows 0.0175. In radian mode it shows 0.8415. The number 1 means 1° in one mode and 1 radian, about 57.3°, in the other. Type sin 30: degree mode gives 0.5 and radian mode gives −0.988.

What is true: Set the calculator to radian mode (often shown as RAD or R) whenever the angle is in radians. Check with sin(π/6): it must show 0.5. If it shows 0.009, the calculator is still in degree mode.

Key words

radian
the angle at the centre of a circle that cuts off an arc equal in length to the radius. About 57.3°.
arc
a part of the edge of a circle.
sector
the slice of a circle between two radii and an arc.
segment
the part of a circle between a chord and the arc it cuts off.
chord
a straight line joining two points on a circle.

Back to your guess

The answer is C: An arc as long as the radius cuts an angle of about 57° at the centre.

The angle is about 57.3°. The whole circle has an arc of 2π radii, which is a little over 6 radii, and 360° ÷ 6.28 is about 57.3°. This angle is called one radian, and it is the natural unit for angles.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is 72° in radians?

    Tap your answer.

    1. Yes, that is right. Multiply by π/180. 72π/180 cancels by 36 to 2π/5. Check: 2π/5 is 2/5 of 180°, and 2/5 of 180 is 72.

    2. Not this one. You turned the fraction upside down. To change degrees into radians, multiply by π/180, so 72° is 72π/180. Go to M5.3.1. Try another.

    3. Not this one. That is 36°. The fraction is 72/180 = 2/5, so the angle is 2/5 of π, which is 2π/5. Go to M5.3.1. Try another.

    4. Not this one. The fraction 72/180 = 0.4 is the part of a half turn. To get the angle in radians, multiply it by π: 0.4π, which is 2π/5. Go to M5.3.1. Try another.

    See the answer

    2π/5

    Multiply by π/180. 72π/180 cancels by 36 to 2π/5. Check: 2π/5 is 2/5 of 180°, and 2/5 of 180 is 72.

    • If you chose B: You turned the fraction upside down. To change degrees into radians, multiply by π/180, so 72° is 72π/180. Go to M5.3.1.
    • If you chose C: That is 36°. The fraction is 72/180 = 2/5, so the angle is 2/5 of π, which is 2π/5. Go to M5.3.1.
    • If you chose D: The fraction 72/180 = 0.4 is the part of a half turn. To get the angle in radians, multiply it by π: 0.4π, which is 2π/5. Go to M5.3.1.

  2. A sector has radius 5 cm and angle 0.8 radians. What is its arc length?

    Tap your answer.

    1. Not this one. You divided the radius by the angle. The arc is s = rθ, so you multiply them. Go to M5.3.1. Try another.

    2. Yes, that is right. The angle is in radians, so s = rθ = 5 × 0.8 = 4 cm.

    3. Not this one. You treated 0.8 as degrees. In radians the angle is already 0.8, so s = 5 × 0.8. Go to M5.3.1. Try another.

    See the answer

    4 cm

    The angle is in radians, so s = rθ = 5 × 0.8 = 4 cm.

    • If you chose A: You divided the radius by the angle. The arc is s = rθ, so you multiply them. Go to M5.3.1.
    • If you chose C: You treated 0.8 as degrees. In radians the angle is already 0.8, so s = 5 × 0.8. Go to M5.3.1.

  3. Which of these angles is the biggest?

    Tap your answer.

    1. Not this one. The number 50 looks large, but 1 radian is 57.3°, which is more than 50°. Go to M5.3.1. Try another.

    2. Not this one. Half a radian is about 28.6°, which is the smallest of the four. Go to M5.3.1. Try another.

    3. Not this one. π/4 radians is 180° ÷ 4 = 45°, which is smaller than 57.3°. Go to M5.3.1. Try another.

    4. Yes, that is right. Put them all in degrees. 1 radian is 57.3°, 50° is 50°, π/4 radians is 45° and 0.5 radian is 28.6°. The biggest is 1 radian.

    See the answer

    1 radian

    Put them all in degrees. 1 radian is 57.3°, 50° is 50°, π/4 radians is 45° and 0.5 radian is 28.6°. The biggest is 1 radian.

    • If you chose A: The number 50 looks large, but 1 radian is 57.3°, which is more than 50°. Go to M5.3.1.
    • If you chose B: Half a radian is about 28.6°, which is the smallest of the four. Go to M5.3.1.
    • If you chose C: π/4 radians is 180° ÷ 4 = 45°, which is smaller than 57.3°. Go to M5.3.1.

  4. Mei works out the area of a sector with r = 4 cm and θ = 60° as ½ × 4² × 60 = 480 cm². What went wrong?

    Tap your answer.

    1. Not this one. That is a way to work in degrees, with (θ/360) × πr². It does not repair ½r²θ, which needs θ in radians. Go to M5.3.1. Try another.

    2. Not this one. The formula ½r²θ is right. The area depends on the square of the radius, since doubling the radius makes the sector four times as big. Go to M5.3.1. Try another.

    3. Yes, that is right. A = ½r²θ needs θ in radians. Here θ = 60° = π/3, so A = ½ × 16 × π/3 = 8π/3, about 8.38 cm². The answer 480 cm² is far too large: the whole circle has an area of only 50.3 cm².

    See the answer

    The sector area formula was used with the angle in degrees.

    A = ½r²θ needs θ in radians. Here θ = 60° = π/3, so A = ½ × 16 × π/3 = 8π/3, about 8.38 cm². The answer 480 cm² is far too large: the whole circle has an area of only 50.3 cm².

    • If you chose A: That is a way to work in degrees, with (θ/360) × πr². It does not repair ½r²θ, which needs θ in radians. Go to M5.3.1.
    • If you chose B: The formula ½r²θ is right. The area depends on the square of the radius, since doubling the radius makes the sector four times as big. Go to M5.3.1.

  5. A calculator shows sin(π/6) = 0.00914. What is the likely cause?

    Tap your answer.

    1. Yes, that is right. In degree mode the calculator reads π/6 = 0.5236 as 0.5236 degrees, and sin 0.5236° = 0.00914. Switch to radian mode and sin(π/6) = 0.5.

    2. Not this one. Radian mode is the correct mode for π/6. In radian mode sin(π/6) = 0.5. Go to M5.3.1. Try another.

    3. Not this one. sin(π/6) = sin 30° = 0.5 exactly. A different value means the mode is wrong. Go to M5.3.1. Try another.

    See the answer

    It is in degree mode.

    In degree mode the calculator reads π/6 = 0.5236 as 0.5236 degrees, and sin 0.5236° = 0.00914. Switch to radian mode and sin(π/6) = 0.5.

    • If you chose B: Radian mode is the correct mode for π/6. In radian mode sin(π/6) = 0.5. Go to M5.3.1.
    • If you chose C: sin(π/6) = sin 30° = 0.5 exactly. A different value means the mode is wrong. Go to M5.3.1.

  6. Change 5π/4 radians to degrees.

    See the answer

    225°

    Multiply by 180/π. The π cancels, leaving 5 × 180 ÷ 4 = 5 × 45 = 225.

  7. A sector has radius 12 cm and angle 2.5 radians. Find its area in cm².

    See the answer

    180

    A = ½r²θ = ½ × 12² × 2.5 = ½ × 144 × 2.5 = 72 × 2.5 = 180 cm².

  8. A sector of a circle has radius 7 cm and perimeter 25 cm. Show that its angle is 11/7 radians, and find its area in cm².

    A sector of a circle with its centre at the bottom. The radius is labelled 7 cm, the arc is labelled s and the angle at the centre is labelled θ. The sector is hatched.θ7 cms
    See the answer

    38.5

    Perimeter 25 = 7 + 7 + s, so s = 11 cm. Then θ = s ÷ r = 11/7, as asked. The area is A = ½rs = ½ × 7 × 11 = 38.5 cm². Check: ½r²θ = ½ × 49 × 11/7 = 38.5.

  9. A chord AB of a circle with radius 12 cm cuts an angle of 0.9 radians at the centre. Find the area of the segment between the chord and the minor arc, in cm² to 2 significant figures.

    See the answer

    8.4

    Segment = ½r²(θ − sin θ). Here ½ × 144 = 72. In radian mode sin 0.9 = 0.78333, so θ − sin θ = 0.9 − 0.78333 = 0.11667. Then 72 × 0.11667 = 8.40, so the area is 8.4 cm².

  10. What is the perimeter of a sector with radius 3 m and angle 2 radians?

    Tap your answer.

    1. Not this one. That is only the arc, rθ = 3 × 2. The perimeter also has the two straight sides. Go to M5.3.1. Try another.

    2. Yes, that is right. The arc is rθ = 3 × 2 = 6 m. Add the two radii, 3 + 3 = 6 m. The perimeter is 6 + 6 = 12 m.

    3. Not this one. You added one radius to the arc. A sector has two straight sides, each 3 m. Go to M5.3.1. Try another.

    See the answer

    12 m

    The arc is rθ = 3 × 2 = 6 m. Add the two radii, 3 + 3 = 6 m. The perimeter is 6 + 6 = 12 m.

    • If you chose A: That is only the arc, rθ = 3 × 2. The perimeter also has the two straight sides. Go to M5.3.1.
    • If you chose C: You added one radius to the arc. A sector has two straight sides, each 3 m. Go to M5.3.1.

  11. For small θ in radians, (1 − cos 2θ) ÷ (θ tan θ) is close to a whole number. Which number?

    See the answer

    2

    For small θ, cos 2θ ≈ 1 − (2θ)²/2 = 1 − 2θ², so 1 − cos 2θ ≈ 2θ². Also tan θ ≈ θ, so θ tan θ ≈ θ². The ratio is close to 2θ² ÷ θ² = 2.

Keep it

Copy this into your exercise book.

π rad = 180°. Degrees to radians: × π/180. Radians to degrees: × 180/π. s = rθ, A = ½r²θ, perimeter 2r + rθ. Radians only. Calculator in RAD mode.

Draw: A sector with radius r, arc s and angle θ, with the three formulas beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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