Chapter 5 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

M4.3.5 Chapter 5

Finding angles

A ramp rises 1 m while it goes 4 m along the ground. What angle does it make with the ground?

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

  • Use inverse sin, cos or tan to find an angle
  • Find the angle of a slope from its height and length

In the last chapter the angle was given and a side was missing. Now it is the other way round. You know two sides and you want the angle.

You already know sin 30° = 0.5. Turn it round. Which angle has a sine of 0.5? The answer is 30°. The key that does this is the inverse sine, written sin⁻¹. On most calculators you press shift, then sin.

A table with the head Angle, sin, cos and tan. Row 1: 15°, 0.26, 0.97, 0.27. Row 2: 30°, 0.5, 0.87, 0.58. Row 3: 45°, 0.71, 0.71, 1. Row 4: 60°, 0.87, 0.5, 1.73. Row 5: 75°, 0.97, 0.26, 3.73. The values are rounded to 2 decimal places.Anglesincostan15°0.260.970.2730°0.50.870.5845°0.710.71160°0.870.51.7375°0.970.263.73
Reading the table from the right gives the inverse: which angle has sin 0.5? The row 30°.

Look at the table. As the angle grows, sin grows from 0 up to 1. A sine of 0.5 sits at 30°, and a sine of 0.87 at 60°. The inverse keys do the same search for any number, not only the ones in a small table.

Take care with the name. sin⁻¹ does not mean 1 ÷ sin. The small −1 says "go backwards". You find an angle from a ratio, and the answer is an angle in degrees. Check that the calculator is in degrees mode.

The three inverse keys follow the same rule as before. Choose the ratio from the two sides you know. Divide them. Then press the inverse key for that ratio.

A sine or cosine in a right angled triangle is always between 0 and 1, because the hypotenuse is the longest side. If your ratio is above 1, the sides are the wrong way round. The calculator shows an error for sin⁻¹ or cos⁻¹ of a number above 1. A tangent can be any number above 0.

A very flat right angled triangle for a ramp. The upright side is labelled 0.5 m, the flat side is labelled 6 m and the angle θ is marked at the right corner.0.5 m6 mθ

Worked example

A ramp rises 0.5 m over a flat distance of 6 m. What angle does it make with the ground? Give the answer to 1 decimal place.

  1. The rise 0.5 m is opposite θ. The flat 6 m is adjacent. We have O and A, so use tan.
  2. tan θ = 0.5 ÷ 6.
  3. Use the inverse tan key on 0.5 ÷ 6, without rounding the division first.
  4. To 4 decimal places, θ = 4.7636.
  5. Round to 1 decimal place: θ = 4.8°.

Answer: To 1 decimal place, the angle is 4.8°.

Now a road. A road is 150 m long and climbs 20 m. The road is the hypotenuse, so the angle uses sine.

A right angled triangle for a hill road. The slanted side, the road, is labelled 150 m. The upright side is labelled 20 m. The angle θ is marked at the right corner.20 m150 mθ

Worked example

A road is 150 m long and climbs 20 m. What angle does the road make with the flat? Give the answer to 1 decimal place.

  1. The climb 20 m is opposite θ. The road, 150 m, is the hypotenuse. We have O and H, so use sin.
  2. To 4 decimal places, sin θ = 20 ÷ 150 = 0.1333.
  3. Use the inverse sin key on 20 ÷ 150.
  4. To 4 decimal places, θ = 7.6623.
  5. Round to 1 decimal place: θ = 7.7°.

Answer: To 1 decimal place, the angle is 7.7°.

Last, a ladder. Its foot is 1.8 m from the wall and it is 6 m long. What angle does it make with the ground? Build the picture in three moves.

Draw it first

Find the angle of a ladder

  1. Step 1: a right angled triangle for a ladder. Step 2: the flat side is labelled 1.8 m and the slanted side, the ladder, is labelled 6 m. Step 3: the angle θ is marked at the left corner, where the ladder meets the ground.

    1 Draw the ladder, the ground and the wall as a right angled triangle.

  2. Step 1: a right angled triangle for a ladder. Step 2: the flat side is labelled 1.8 m and the slanted side, the ladder, is labelled 6 m. Step 3: the angle θ is marked at the left corner, where the ladder meets the ground.1.8 m6 m

    2 Write the known lengths: the foot distance 1.8 m and the ladder 6 m.

  3. Step 1: a right angled triangle for a ladder. Step 2: the flat side is labelled 1.8 m and the slanted side, the ladder, is labelled 6 m. Step 3: the angle θ is marked at the left corner, where the ladder meets the ground.1.8 m6 mθ

    3 Mark θ where the ladder meets the ground.

Worked example

A ladder 6 m long has its foot 1.8 m from a wall. What angle does it make with the ground? Give the answer to 1 decimal place.

  1. The foot distance 1.8 m is adjacent to θ. The ladder, 6 m, is the hypotenuse. We have A and H, so use cos.
  2. cos θ = 1.8 ÷ 6 = 0.3.
  3. Use the inverse cos key on 0.3.
  4. To 4 decimal places, θ = 72.5424.
  5. Round to 1 decimal place: θ = 72.5°.

Answer: To 1 decimal place, the angle is 72.5°.

See it three ways

With things

A road climbs 20 m for every 150 m you walk along it.

As a picture

A right angled triangle for a hill road. The slanted side is labelled 150 m, the upright side is labelled 20 m and the angle θ is marked at the right corner.20 m150 mθ

In symbols

sin θ = 20 ÷ 150, so θ = sin⁻¹(20 ÷ 150), which is 7.7° to 1 decimal place.

Try it

Find the angle of a ruler

You need: a ruler 30 cm long, a pile of books, a protractor.

  1. Make a pile of books 12 cm high. Lean the ruler from the floor to the top of the pile.
  2. The ruler is 30 cm long and the pile is 12 cm high. Work out sin⁻¹(12 ÷ 30).
  3. Now measure the angle between the ruler and the floor with the protractor.

What you will see: The inverse sine gives 23.6°, and the protractor should show about the same.

If you cannot try it, this is what happens.
A right angled triangle for a ruler leaning on books. The slanted side, the ruler, is labelled 30 cm. The upright side is labelled 12 cm. The angle of 23.6° is marked at the left corner.12 cm30 cm23.6°

A wrong idea many people have

The ratio is the angle: tan θ = 0.25 means θ = 25°.

Why it feels right: The digits 25 match, and a calculator shows a number, so the number seems to be the answer.

The test that breaks it: Work out tan 25° on the calculator. It gives 0.466, not 0.25. The angle that has tan 0.25 is 14.04°.

What is true: A ratio is a number about the shape, and an angle is a turn. Use the inverse key to change one into the other.

A wrong idea many people have

sin⁻¹ 0.5 means 1 ÷ sin 0.5.

Why it feels right: In algebra a small −1 above a number means the reciprocal: 2⁻¹ = 1 ÷ 2.

The test that breaks it: We know sin 30° = 0.5. If sin⁻¹ 0.5 were 1 ÷ sin 0.5, the answer would not be 30. The inverse key gives 30°, so it goes backwards, it does not divide.

What is true: On a function key, ⁻¹ means the inverse function. sin⁻¹ undoes sin.

Key words

inverse sine
sin⁻¹: the angle whose sine is a given ratio. Likewise cos⁻¹ and tan⁻¹.

Back to your guess

The answer is A: About 14°, found with the inverse tangent of 1 ÷ 4.

The angle is about 14°. The ratio 0.25 is not the angle. It is the tangent of the angle, and the inverse tan key on a calculator turns it back into 14.04°. A ramp at 45° would rise 4 m over 4 m, not 1 m.

Check yourself

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

  1. A right angled triangle has opposite 5 cm and hypotenuse 8 cm for the angle θ. Which sum gives θ?

    Tap your answer.

    1. Yes, that is right. The 5 cm side is opposite θ and the 8 cm side is the hypotenuse. That is O and H, so the ratio is sin. Divide, then press the inverse sin key.

    2. Not this one. Cos joins the adjacent side and the hypotenuse. You were given the opposite side, so the ratio must be sin. Go to M4.3.4. Try another.

    3. Not this one. Tan joins the opposite and the adjacent side. The 8 cm is the hypotenuse, so the ratio must be sin. Go to M4.3.4. Try another.

    4. Not this one. That is the ratio, 0.625, not the angle. Press the inverse sin key to turn the ratio into an angle. Go to M4.3.5. Try another.

    See the answer

    θ = inverse sin of 5 ÷ 8

    The 5 cm side is opposite θ and the 8 cm side is the hypotenuse. That is O and H, so the ratio is sin. Divide, then press the inverse sin key.

    • If you chose B: Cos joins the adjacent side and the hypotenuse. You were given the opposite side, so the ratio must be sin. Go to M4.3.4.
    • If you chose C: Tan joins the opposite and the adjacent side. The 8 cm is the hypotenuse, so the ratio must be sin. Go to M4.3.4.
    • If you chose D: That is the ratio, 0.625, not the angle. Press the inverse sin key to turn the ratio into an angle. Go to M4.3.5.

  2. A ramp rises 1.2 m over a flat distance of 5 m. What angle does it make with the ground, to 1 decimal place?

    A flat right angled triangle for a ramp. The upright side is labelled 1.2 m, the flat side is labelled 5 m and the angle θ is marked at the right corner.1.2 m5 mθ
    Tap your answer.

    1. Not this one. You divided the adjacent side by the opposite, 5 ÷ 1.2. That gives the other angle of the triangle, the one at the top of the ramp. Put the opposite side on top. Go to M4.3.5. Try another.

    2. Yes, that is right. tan θ = 1.2 ÷ 5, so θ = tan⁻¹(0.24) = 13.4957, which is 13.5° to 1 decimal place.

    3. Not this one. You used sin⁻¹ of 1.2 ÷ 5. The 5 m is the flat side, the adjacent, not the hypotenuse. The right ratio is tan. Go to M4.3.4. Try another.

    4. Not this one. You stopped at the ratio, 1.2 ÷ 5 = 0.24. That is tan θ. Press the inverse tan key to get the angle. Go to M4.3.5. Try another.

    See the answer

    13.5°

    tan θ = 1.2 ÷ 5, so θ = tan⁻¹(0.24) = 13.4957, which is 13.5° to 1 decimal place.

    • If you chose A: You divided the adjacent side by the opposite, 5 ÷ 1.2. That gives the other angle of the triangle, the one at the top of the ramp. Put the opposite side on top. Go to M4.3.5.
    • If you chose C: You used sin⁻¹ of 1.2 ÷ 5. The 5 m is the flat side, the adjacent, not the hypotenuse. The right ratio is tan. Go to M4.3.4.
    • If you chose D: You stopped at the ratio, 1.2 ÷ 5 = 0.24. That is tan θ. Press the inverse tan key to get the angle. Go to M4.3.5.

  3. A ladder 7 m long has its foot 2 m from a wall. What angle does it make with the ground, to 1 decimal place?

    A right angled triangle for a ladder. The flat side is labelled 2 m, the slanted side, the ladder, is labelled 7 m and the angle θ is marked at the left corner, where the ladder meets the ground.2 m7 mθ
    Tap your answer.

    1. Not this one. You used sin⁻¹ of 2 ÷ 7. That is the angle between the ladder and the wall, at the top. The angle with the ground is the other one. Go to M4.3.5. Try another.

    2. Yes, that is right. The foot 2 m is adjacent to θ and the ladder is the hypotenuse, so cos θ = 2 ÷ 7. Then θ = cos⁻¹(0.2857) = 73.3985, which is 73.4° to 1 decimal place.

    3. Not this one. You used tan⁻¹ of 2 ÷ 7. But 2 m is adjacent and 7 m is the hypotenuse, not the opposite side. The ratio with A and H is cos. Go to M4.3.4. Try another.

    4. Not this one. You divided 2 by 7 and read the answer 0.286 as 28.6°. The division gives a ratio, not an angle. Press the inverse cos key. Go to M4.3.5. Try another.

    See the answer

    73.4°

    The foot 2 m is adjacent to θ and the ladder is the hypotenuse, so cos θ = 2 ÷ 7. Then θ = cos⁻¹(0.2857) = 73.3985, which is 73.4° to 1 decimal place.

    • If you chose A: You used sin⁻¹ of 2 ÷ 7. That is the angle between the ladder and the wall, at the top. The angle with the ground is the other one. Go to M4.3.5.
    • If you chose C: You used tan⁻¹ of 2 ÷ 7. But 2 m is adjacent and 7 m is the hypotenuse, not the opposite side. The ratio with A and H is cos. Go to M4.3.4.
    • If you chose D: You divided 2 by 7 and read the answer 0.286 as 28.6°. The division gives a ratio, not an angle. Press the inverse cos key. Go to M4.3.5.

  4. On a calculator, sin⁻¹ of a number gives an error. The number came from dividing two sides of a right angled triangle. What is the most likely reason?

    Tap your answer.

    1. Yes, that is right. The sine of an angle in a right angled triangle is opposite ÷ hypotenuse. The hypotenuse is the longest side, so the answer is below 1. A number above 1 means the sides were swapped.

    2. Not this one. It is more likely to be the sum. A sine in a right angled triangle cannot be above 1, because the hypotenuse is the longest side. Check the division. Go to M4.3.5. Try another.

    3. Not this one. The other two angles of a right angled triangle are both less than 90°. The error comes from the ratio being above 1, not from the angle. Go to M4.3.5. Try another.

    See the answer

    The sides were divided the wrong way round, so the ratio was more than one.

    The sine of an angle in a right angled triangle is opposite ÷ hypotenuse. The hypotenuse is the longest side, so the answer is below 1. A number above 1 means the sides were swapped.

    • If you chose B: It is more likely to be the sum. A sine in a right angled triangle cannot be above 1, because the hypotenuse is the longest side. Check the division. Go to M4.3.5.
    • If you chose C: The other two angles of a right angled triangle are both less than 90°. The error comes from the ratio being above 1, not from the angle. Go to M4.3.5.

  5. A steep path is 15 m long and climbs 9 m. What angle does it make with the flat, to 1 decimal place?

    A right angled triangle for a steep path. The slanted side, the path, is labelled 15 m. The upright side is labelled 9 m. The angle θ is marked at the right corner.9 m15 mθ
    Tap your answer.

    1. Not this one. You used cos⁻¹ of 9 ÷ 15. But 9 m is the opposite side, not the adjacent. That gives the other angle of the triangle. The ratio with O and H is sin. Go to M4.3.4. Try another.

    2. Not this one. You used tan⁻¹ of 9 ÷ 15. Tan needs the opposite and the adjacent. The 15 m is the hypotenuse, so the ratio is sin. Go to M4.3.4. Try another.

    3. Yes, that is right. sin θ = 9 ÷ 15 = 0.6, so θ = sin⁻¹(0.6) = 36.8699, which is 36.9° to 1 decimal place.

    4. Not this one. You stopped at the ratio, 9 ÷ 15 = 0.6. That is sin θ. Press the inverse sin key to get the angle. Go to M4.3.5. Try another.

    See the answer

    36.9°

    sin θ = 9 ÷ 15 = 0.6, so θ = sin⁻¹(0.6) = 36.8699, which is 36.9° to 1 decimal place.

    • If you chose A: You used cos⁻¹ of 9 ÷ 15. But 9 m is the opposite side, not the adjacent. That gives the other angle of the triangle. The ratio with O and H is sin. Go to M4.3.4.
    • If you chose B: You used tan⁻¹ of 9 ÷ 15. Tan needs the opposite and the adjacent. The 15 m is the hypotenuse, so the ratio is sin. Go to M4.3.4.
    • If you chose D: You stopped at the ratio, 9 ÷ 15 = 0.6. That is sin θ. Press the inverse sin key to get the angle. Go to M4.3.5.

  6. From an earlier chapter

    A ramp 8 m long slopes at 15° to the ground. How high does it rise, to 1 decimal place?

    Tap your answer.

    1. Not this one. You used cos. Cos gives the flat distance, the adjacent side. The height is opposite the angle, so use sin. Go to M4.3.4. Try another.

    2. Not this one. You divided 8 by sin 15°. The height is on top of the ratio, so multiply: height = 8 × sin 15°. A height of 30.9 m is longer than the ramp, which cannot be. Go to M4.3.4. Try another.

    3. Not this one. Your calculator was in radians mode. It worked out sin of 15 radians. Switch to degrees mode. Go to M4.3.4. Try another.

    4. Yes, that is right. sin 15° = height ÷ 8, so height = 8 × sin 15° = 2.071, which is 2.1 m to 1 decimal place.

    See the answer

    2.1 m

    sin 15° = height ÷ 8, so height = 8 × sin 15° = 2.071, which is 2.1 m to 1 decimal place.

    • If you chose A: You used cos. Cos gives the flat distance, the adjacent side. The height is opposite the angle, so use sin. Go to M4.3.4.
    • If you chose B: You divided 8 by sin 15°. The height is on top of the ratio, so multiply: height = 8 × sin 15°. A height of 30.9 m is longer than the ramp, which cannot be. Go to M4.3.4.
    • If you chose C: Your calculator was in radians mode. It worked out sin of 15 radians. Switch to degrees mode. Go to M4.3.4.

  7. A ramp rises 2 m over a flat distance of 9 m. What is the angle with the ground, in degrees, to 1 decimal place?

    See the answer

    12.5

    tan θ = 2 ÷ 9, so θ = tan⁻¹(0.2222) = 12.5288. To 1 decimal place that is 12.5.

  8. A ladder 9 m long has its foot 2.4 m from a wall. What angle does it make with the ground, in degrees, to 1 decimal place?

    See the answer

    74.5

    cos θ = 2.4 ÷ 9 = 0.2667, so θ = cos⁻¹(0.2667) = 74.5340, which is 74.5 to 1 decimal place.

Keep it

Copy this into your exercise book.

Two sides known, angle wanted. Pick sin, cos or tan. Divide the sides, then press the inverse key. Use degrees mode.

Draw: A right angled triangle with two sides labelled, θ marked, and θ = tan⁻¹(opposite ÷ adjacent) 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.