Chapter 4 of 8

All 8 chapters
  1. 1Pythagoras' theorem, readunderstood
  2. 2Using Pythagoras, readunderstood
  3. 3Opposite, adjacent and hypotenuse, readunderstood
  4. 4Sine, cosine and tangent, readunderstood
  5. 5Finding angles, readunderstood
  6. 6Exact values, readunderstood
  7. 7The sine rule and the cosine rule, readunderstood
  8. 8Area of any triangle, and bearings, readunderstood

M4.3.4 Chapter 4

Sine, cosine and tangent

A ladder 5 m long leans on a wall at 75° to the ground. How high up the wall does it reach?

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

  • Choose sin, cos or tan from the sides you know
  • Use sin, cos or tan to find a missing side

In the last chapter you saw that opposite ÷ adjacent is fixed for an angle θ. That fixed number has a name. It is the tangent of θ, written tan θ.

There are two more fixed numbers, and the same argument shows they are fixed too. The sine of θ is opposite ÷ hypotenuse, written sin θ. The cosine of θ is adjacent ÷ hypotenuse, written cos θ.

A right angled triangle ABC with a small square at C and the angle θ marked at A. The side BC is labelled opposite 3, the side AC is labelled adjacent 4 and the side AB is labelled hypotenuse 5.ABCopposite 3adjacent 4hypotenuse 5θ
The sides are 3, 4 and 5 for this θ.
A table with the head Ratio, Sides and Value. Row 1: sin θ, 3 ÷ 5, 0.6. Row 2: cos θ, 4 ÷ 5, 0.8. Row 3: tan θ, 3 ÷ 4, 0.75.RatioSidesValuesin θ3 ÷ 50.6cos θ4 ÷ 50.8tan θ3 ÷ 40.75
Three ratios of the same triangle. Each is a fixed number for this θ.

For this triangle θ is about 36.9°. Put your calculator in degrees mode, often shown as DEG. Then sin 36.87° gives about 0.6, cos 36.87° gives about 0.8 and tan 36.87° gives about 0.75. The calculator holds a table of these numbers for every angle.

To remember the three ratios, use SOH CAH TOA. SOH: Sine is Opposite over Hypotenuse. CAH: Cosine is Adjacent over Hypotenuse. TOA: Tangent is Opposite over Adjacent.

A table with the head Name and Ratio. Row 1: SOH, sin = O ÷ H. Row 2: CAH, cos = A ÷ H. Row 3: TOA, tan = O ÷ A. Here O is opposite, A is adjacent and H is hypotenuse.NameRatioSOHsin = O ÷ HCAHcos = A ÷ HTOAtan = O ÷ A

To find a missing side, do three things. Label the sides O, A and H for your angle. Find the two sides that matter: the one you know and the one you want. Pick the ratio that has exactly those two letters.

Start with the ladder from the question. It makes a right angled triangle with the wall and the ground. The ladder is H, 5 m. The height h is opposite the 75° angle. The two letters are O and H, so you use sin.

A right angled triangle with an angle of 75° at the left corner. The ladder, the slanted side, is labelled 5 m and the upright side is labelled h.h5 m75°

Worked example

A ladder 5 m long leans on a wall at 75° to the ground. How high up the wall does it reach? Give the answer to 1 decimal place.

  1. The ladder is the hypotenuse. The height h is opposite the 75° angle. We have O and H, so use sin.
  2. sin 75° = h ÷ 5, so h = 5 × sin 75°.
  3. On the calculator, to 4 decimal places, sin 75° = 0.9659.
  4. Multiply: 5 × 0.9659 = 4.8295.
  5. Round to 1 decimal place: h = 4.8 m.

Answer: To 1 decimal place, the ladder reaches 4.8 m.

Now a case with cosine. A ramp is 6 m long and slopes at 20° to the flat ground. How far does it stretch along the ground? Draw the triangle first.

Draw it first

Choose the ratio for a ramp

  1. Step 1: a right angled triangle with a small square at the left corner and an angle of 20° at the right corner. Step 2: the sides are labelled. The slanted side, the ramp, is labelled 6 m and the flat side is labelled x.20°

    1 Draw the ramp as the slanted side. Mark the 20° angle with the ground.

  2. Step 1: a right angled triangle with a small square at the left corner and an angle of 20° at the right corner. Step 2: the sides are labelled. The slanted side, the ramp, is labelled 6 m and the flat side is labelled x.20°x6 m

    2 Label the ramp 6 m, the hypotenuse. Label the flat distance x. It is adjacent to the 20° angle. We have A and H, so use cos.

Worked example

A ramp 6 m long slopes at 20° to flat ground. How far does it stretch along the ground? Give the answer to 1 decimal place.

  1. The ramp is the hypotenuse. The distance x is adjacent to the 20° angle. We have A and H, so use cos.
  2. cos 20° = x ÷ 6, so x = 6 × cos 20°.
  3. On the calculator, to 4 decimal places, cos 20° = 0.9397.
  4. Multiply: 6 × 0.9397 = 5.6382.
  5. Round to 1 decimal place: x = 5.6 m.

Answer: To 1 decimal place, the ramp stretches 5.6 m.

Sometimes the side you want is on the bottom of the ratio. Then you divide. A tree is 12 m tall. From a point on level ground, the top is seen at an angle of 35° above the ground. How far is the point from the foot of the tree?

A right angled triangle with an angle of 35° at the right corner. The upright side is labelled 12 m and the flat side is labelled x.12 mx35°

Worked example

The top of a tree 12 m tall is seen at 35° above the ground. How far is the viewer from the foot of the tree? Give the answer to 1 decimal place.

  1. The 12 m height is opposite the 35° angle. The distance x is adjacent. We have O and A, so use tan.
  2. tan 35° = 12 ÷ x, so x × tan 35° = 12.
  3. Divide both sides by tan 35°: x = 12 ÷ tan 35°.
  4. On the calculator, to 4 decimal places, tan 35° = 0.7002.
  5. Divide, to 3 decimal places: 12 ÷ 0.7002 = 17.138.
  6. Round to 1 decimal place: x = 17.1 m.

Answer: To 1 decimal place, the viewer is 17.1 m from the tree.

Keep the rule: when the unknown is on top of the ratio, multiply. When it is on the bottom, divide. In real work keep all the digits on your calculator and round only at the end. In the three examples we rounded the ratio to 4 places only to show each step. The final answers are the same.

See it three ways

With things

A ramp is 5 m long and rises 3 m. For every metre of ramp, you rise 3 ÷ 5 = 0.6 m.

As a picture

A right angled triangle with a small square at C and the angle θ marked at A. The side BC is labelled rise 3 m and the side AB is labelled ramp 5 m.ABCrise 3 mramp 5 mθ

In symbols

sin θ = opposite ÷ hypotenuse = 3 ÷ 5 = 0.6.

Try it

Measure a sine

You need: paper, a ruler, a protractor, a pencil.

  1. Draw a flat line. At its left end use the protractor to draw a line at 30° above it, 10 cm long. This is the hypotenuse.
  2. From the top end of the slanted line, draw a line straight down to the flat line. Measure it.
  3. Divide the measured length by 10.

What you will see: The vertical line is about 5 cm, so the sine is 5 ÷ 10 = 0.5. Your calculator says sin 30° = 0.5. The paper and the calculator agree.

If you cannot try it, this is what happens.
A right angled triangle with an angle of 30° at the left corner. The slanted side is labelled 10 cm, the upright side is labelled 5 cm and the flat side is labelled 8.7 cm.5 cm8.7 cm10 cm30°

A wrong idea many people have

On my calculator sin 30° is −0.988, so the calculator has a fault.

Why it feels right: A calculator is exact, so a strange answer must be a broken button.

The test that breaks it: Look for DEG or D on the screen. Some calculators start in radians. In radians mode, sin 30 means 30 radians, and that gives −0.988.

What is true: Set the calculator to degrees. Then sin 30° = 0.5, which matches the drawing.

A wrong idea many people have

If sin 30° = 0.5, then sin 60° = 1, because 60° is twice 30°.

Why it feels right: Doubling the angle looks like it should double the answer.

The test that breaks it: Draw a 60° angle with a hypotenuse of 10 cm. The upright side is about 8.7 cm, so sin 60° is about 0.87, not 1.

What is true: The ratios do not grow in step with the angle. Use the calculator for each angle.

Key words

sine
sin θ = opposite ÷ hypotenuse.
cosine
cos θ = adjacent ÷ hypotenuse.
tangent
tan θ = opposite ÷ adjacent.

Back to your guess

The answer is C: About 4.8 m, because the ladder is nearly upright.

The ladder reaches about 4.8 m. It cannot reach 5 m, because the ladder is the hypotenuse and the height is a shorter side. A percentage of the angle is not a length. You find the height with 5 × sin 75°, and this chapter shows how.

Check yourself

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

  1. You know the hypotenuse, 12 cm, and the angle, 40°. You want the opposite side x. Which ratio do you use?

    A right angled triangle with an angle of 40° at the right corner. The slanted side is labelled 12 cm and the upright side is labelled x.x12 cm40°
    Tap your answer.

    1. Not this one. Cos uses the adjacent side. The side x faces the angle, so it is the opposite side, not the adjacent one. Go to M4.3.3. Try another.

    2. Yes, that is right. x faces the angle, so it is O. The 12 cm side is the hypotenuse, H. The ratio with O and H is sin, from SOH.

    3. Not this one. Tan needs the adjacent side, and here you do not know it. You know the hypotenuse, so the ratio must have H in it. Go to M4.3.4. Try another.

    See the answer

    sin, because it joins opposite and hypotenuse

    x faces the angle, so it is O. The 12 cm side is the hypotenuse, H. The ratio with O and H is sin, from SOH.

    • If you chose A: Cos uses the adjacent side. The side x faces the angle, so it is the opposite side, not the adjacent one. Go to M4.3.3.
    • If you chose C: Tan needs the adjacent side, and here you do not know it. You know the hypotenuse, so the ratio must have H in it. Go to M4.3.4.

  2. You know the adjacent side, 9 cm, and the angle, 55°. You want the opposite side x. Which ratio do you use?

    A right angled triangle with an angle of 55° at the right corner. The flat side is labelled 9 cm and the upright side is labelled x.x9 cm55°
    Tap your answer.

    1. Not this one. Sin needs the hypotenuse. In this triangle you know the adjacent side and want the opposite one, so the hypotenuse is not in the sum. Go to M4.3.4. Try another.

    2. Not this one. Cos needs the hypotenuse too, and the side you want is not the adjacent. Pick the ratio with O and A. Go to M4.3.4. Try another.

    3. Yes, that is right. x is O and the 9 cm side is A. The ratio with O and A is tan, from TOA.

    See the answer

    tan, because it joins opposite and adjacent

    x is O and the 9 cm side is A. The ratio with O and A is tan, from TOA.

    • If you chose A: Sin needs the hypotenuse. In this triangle you know the adjacent side and want the opposite one, so the hypotenuse is not in the sum. Go to M4.3.4.
    • If you chose B: Cos needs the hypotenuse too, and the side you want is not the adjacent. Pick the ratio with O and A. Go to M4.3.4.

  3. A ladder 6 m long leans on a wall at 70° to the ground. How high up the wall does it reach, to 1 decimal place?

    A right angled triangle with an angle of 70° at the left corner. The slanted side is labelled 6 m and the upright side is labelled h.h6 m70°
    Tap your answer.

    1. Yes, that is right. h = 6 × sin 70° = 6 × 0.9397 = 5.638, which is 5.6 m to 1 decimal place.

    2. Not this one. You used cos, which gives the adjacent side, the distance along the ground. The height is opposite the angle, so use sin. Go to M4.3.4. Try another.

    3. Not this one. You used tan. That joins opposite and adjacent, and the adjacent side is not given. The ladder is the hypotenuse, so use sin. A height of 16.5 m is far more than the ladder. Go to M4.3.4. Try another.

    4. Not this one. Your calculator was in radians mode. It worked out sin of 70 radians. Switch to degrees mode and try again. Go to M4.3.4. Try another.

    See the answer

    5.6 m

    h = 6 × sin 70° = 6 × 0.9397 = 5.638, which is 5.6 m to 1 decimal place.

    • If you chose B: You used cos, which gives the adjacent side, the distance along the ground. The height is opposite the angle, so use sin. Go to M4.3.4.
    • If you chose C: You used tan. That joins opposite and adjacent, and the adjacent side is not given. The ladder is the hypotenuse, so use sin. A height of 16.5 m is far more than the ladder. Go to M4.3.4.
    • If you chose D: Your calculator was in radians mode. It worked out sin of 70 radians. Switch to degrees mode and try again. Go to M4.3.4.

  4. A slide is 2.4 m high at its top end. It slopes at 50° to the ground. How long is the slide, to 1 decimal place?

    A right angled triangle with an angle of 50° at the right corner. The upright side is labelled 2.4 m and the slanted side, the slide, is labelled x.2.4 mx50°
    Tap your answer.

    1. Not this one. You multiplied 2.4 by sin 50°. The slide is the hypotenuse, on the bottom of the ratio, so you divide: x = 2.4 ÷ sin 50°. A slide cannot be shorter than its height. Go to M4.3.4. Try another.

    2. Not this one. You divided by cos 50°. Cos joins adjacent and hypotenuse, but the 2.4 m is the opposite side. Use sin. Go to M4.3.4. Try another.

    3. Not this one. You divided by tan 50°. Tan joins opposite and adjacent, and the slide is neither. The slide is the hypotenuse, so use sin. Go to M4.3.4. Try another.

    4. Yes, that is right. sin 50° = 2.4 ÷ x, so x = 2.4 ÷ sin 50° = 2.4 ÷ 0.7660 = 3.133, which is 3.1 m to 1 decimal place.

    See the answer

    3.1 m

    sin 50° = 2.4 ÷ x, so x = 2.4 ÷ sin 50° = 2.4 ÷ 0.7660 = 3.133, which is 3.1 m to 1 decimal place.

    • If you chose A: You multiplied 2.4 by sin 50°. The slide is the hypotenuse, on the bottom of the ratio, so you divide: x = 2.4 ÷ sin 50°. A slide cannot be shorter than its height. Go to M4.3.4.
    • If you chose B: You divided by cos 50°. Cos joins adjacent and hypotenuse, but the 2.4 m is the opposite side. Use sin. Go to M4.3.4.
    • If you chose C: You divided by tan 50°. Tan joins opposite and adjacent, and the slide is neither. The slide is the hypotenuse, so use sin. Go to M4.3.4.

  5. You stand 20 m from the foot of a pole on level ground. You see the top at 25° above the level of your eye. How tall is the pole above your eye level, to 1 decimal place?

    A right angled triangle with an angle of 25° at the right corner. The flat side is labelled 20 m and the upright side is labelled h.h20 m25°
    Tap your answer.

    1. Not this one. You divided 20 by tan 25°. The unknown h is on top of the ratio: tan 25° = h ÷ 20, so you multiply. A pole taller than twice the distance would be at a steep angle. Go to M4.3.4. Try another.

    2. Yes, that is right. tan 25° = h ÷ 20, so h = 20 × tan 25° = 20 × 0.4663 = 9.326, which is 9.3 m to 1 decimal place.

    3. Not this one. You used sin. Sin joins opposite and hypotenuse, and the hypotenuse is not given. The 20 m is the adjacent side, so use tan. Go to M4.3.4. Try another.

    4. Not this one. You used cos. Cos joins adjacent and hypotenuse, but the side you want is the opposite. Use tan. Go to M4.3.4. Try another.

    See the answer

    9.3 m

    tan 25° = h ÷ 20, so h = 20 × tan 25° = 20 × 0.4663 = 9.326, which is 9.3 m to 1 decimal place.

    • If you chose A: You divided 20 by tan 25°. The unknown h is on top of the ratio: tan 25° = h ÷ 20, so you multiply. A pole taller than twice the distance would be at a steep angle. Go to M4.3.4.
    • If you chose C: You used sin. Sin joins opposite and hypotenuse, and the hypotenuse is not given. The 20 m is the adjacent side, so use tan. Go to M4.3.4.
    • If you chose D: You used cos. Cos joins adjacent and hypotenuse, but the side you want is the opposite. Use tan. Go to M4.3.4.

  6. From an earlier chapter

    A small triangle has opposite 4 cm and adjacent 5 cm for θ. A bigger one has the same θ and adjacent 20 cm. How long is its opposite side?

    Tap your answer.

    1. Not this one. You added 15, the growth of the adjacent side, to 4. Sides grow by multiplying: 5 became 20, which is 4 times as much. Go to M4.3.3. Try another.

    2. Not this one. You added 20 and 5. The opposite side keeps the fixed ratio 4 ÷ 5 with the adjacent side, so it must be less than 20. Go to M4.3.3. Try another.

    3. Yes, that is right. The big triangle is 4 times as big, so its opposite is 4 × 4 = 16.

    4. Not this one. You multiplied 4 by 20. The ratio 4 ÷ 5 is fixed, so the opposite is 4 ÷ 5 of 20, and that is less than 20. Go to M4.3.3. Try another.

    See the answer

    16 cm

    The big triangle is 4 times as big, so its opposite is 4 × 4 = 16.

    • If you chose A: You added 15, the growth of the adjacent side, to 4. Sides grow by multiplying: 5 became 20, which is 4 times as much. Go to M4.3.3.
    • If you chose B: You added 20 and 5. The opposite side keeps the fixed ratio 4 ÷ 5 with the adjacent side, so it must be less than 20. Go to M4.3.3.
    • If you chose D: You multiplied 4 by 20. The ratio 4 ÷ 5 is fixed, so the opposite is 4 ÷ 5 of 20, and that is less than 20. Go to M4.3.3.

  7. A ladder 8 m long leans on a wall at 72° to the ground. How many metres up the wall does it reach, to 1 decimal place?

    See the answer

    7.6

    h = 8 × sin 72° = 7.608, which is 7.6 to 1 decimal place.

  8. A ramp rises 1.5 m and makes an angle of 12° with the ground. How many metres long is the ramp, to 1 decimal place?

    See the answer

    7.2

    sin 12° = 1.5 ÷ x, so x = 1.5 ÷ sin 12° = 7.215, which is 7.2 to 1 decimal place.

Keep it

Copy this into your exercise book.

SOH CAH TOA: sin = O ÷ H, cos = A ÷ H, tan = O ÷ A. Label O, A, H. Pick the ratio with your two sides. Unknown on top: multiply. Unknown below: divide.

Draw: A right angled triangle with θ, the sides marked O, A and H, and the words SOH CAH TOA under it.

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

No internet tomorrow? Save the whole book as one file (776 KB). It opens on any phone.