Year 9 · Unit 2 Pythagoras, similarity and trigonometry · Lesson 7 of 7

Working back from the sides to the angle

When two sides are known, a ratio of them pins down the angle. The inverse key on the calculator turns that ratio back into degrees.

Geometry and TrigonometryScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the opposite side.
    Answer5 cm, because 10 × sin 30° = 10 × 0.5.
  2. Opposite 6 and adjacent 8: find tan θ.
    Answer0.75, because 6 ÷ 8.
  3. What operation undoes squaring?
    AnswerTaking the square root.
Words used in this lesson
inverse
the operation that undoes another. For example: tan−1 undoes tan.
angle of elevation
the angle measured up from the horizontal. For example: looking up at a tower.
angle of depression
the angle measured down from the horizontal. For example: looking down at a boat.
horizontal
level, like the surface of still water. For example: the sea.
rounding
giving a number to a stated accuracy. For example: 37.87 is 37.9 to 1 decimal place.

From the sides to the angle

A ramp rises 0.9 m over a horizontal distance of 10.8 m. Is its angle with the ground more than 10° or less? Make a guess and say how you decided before you read on.

To find an angle: label the sides you know (opposite, adjacent, hypotenuse), pick the ratio that uses those two, work out its value, then use the inverse key. If tan θ = 0.75 then θ = tan−1(0.75) = 36.9°.

Elevation and depression: an angle of elevation is measured up from the horizontal to the line of sight; an angle of depression is measured down from the horizontal. Both are measured from the horizontal, so they are angles inside a right-angled triangle.

Set the opposite and adjacent sides. Estimate the angle first, then read it.

5
7
0.714tan θ = opposite ÷ adjacent
35.5°the angle θ
8.6the hypotenuse

tan θ = 5 ÷ 7 = 0.714, so θ = 35.5°. The hypotenuse is √74 ≈ 8.6, and sin and cos give the same angle.

Show me the steps
  1. Opposite 3, adjacent 4: tan θ = 0.75, so θ = 36.9°. The hypotenuse is 5 and sin θ = 0.6 gives the same 36.9°.
  2. Opposite 5, adjacent 5: tan θ = 1, so θ = 45°. The two shorter sides are equal and so are the two base angles.
  3. Opposite 1, adjacent 10: tan θ = 0.1, so θ = 5.7°. A long thin triangle has a small angle.
  4. Opposite 9, adjacent 2: tan θ = 4.5, so θ = 77.5°. A tall thin triangle has a large angle.

Two slips to watch for. tan−1 is not 1 ÷ tan: it is the angle whose tangent is the number you give it. And the angle at the bottom of a tall thin triangle is large, not small, so a tan above 1 should give an angle above 45°.

Worked examples

The ramp: 0.9 m rise over 10.8 m of ground. Find the angle with the ground to 1 decimal place.

  1. The rise is opposite the angle and the ground distance is adjacent, so use tan.
  2. tan θ = 0.9 ÷ 10.8 = 0.0833…
  3. θ = tan−1(0.0833…) = 4.8°, which is less than 10°.

A kite string is 40 m long and the kite is 25 m above the holder's hand. What angle does the string make with the horizontal?

  1. The height 25 m is opposite the angle and the string, 40 m, is the hypotenuse, so use sin.
  2. sin θ = 25 ÷ 40 = 0.625.
  3. θ = sin−1(0.625) = 38.7° to 1 decimal place.

Your go. The first step is given. A rectangle is 9 cm long and 4 cm wide. What angle does a diagonal make with the long side, to 1 decimal place?

  1. The diagonal splits the rectangle into a right-angled triangle with the 4 cm side opposite the angle and the 9 cm side adjacent.
  2. Choose the ratio, work it out, and use the inverse key. Then check.
Check the answer

tan θ = 4 ÷ 9 = 0.444…, so θ = tan−1(0.444…) = 24.0° to 1 decimal place.

With a calculator

To find an angle, use the second function of the key: shift or 2nd, then sin, cos or tan, then the ratio, then equals. For tan−¹ of 0.75 the display should read 36.8698…, which is 36.9°. The key labelled tan−¹ is not “1 ÷ tan”. Keep degree mode on, and round only the final angle.

Check before you go on

1. A right-angled triangle has an opposite side of 4 and a hypotenuse of 9. Which expression gives the angle?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. cos θ = 0.6 for an acute angle θ. Find θ to 1 decimal place.

3. In a right-angled triangle the side opposite θ is 7 and the side adjacent to θ is 9. Find θ to 1 decimal place.

Level 2 · Building

4. A 6 m ladder leans against a wall with its foot 2 m from the wall. What angle does the ladder make with the ground, to 1 decimal place?

5. From a point 45 m from the foot of a tower, the angle of elevation of the top is 32°. How tall is the tower to 1 decimal place?

6. A staircase rises 2.8 m over a horizontal distance of 4.2 m. What angle does it make with the horizontal, to 1 decimal place?

Level 3 · Stretch

7. An isosceles triangle has two equal sides of 10 cm and a base of 8 cm. Find a base angle to 1 decimal place.

8. From the top of a 30 m cliff, the angle of depression of a boat is 18°. How far is the boat from the foot of the cliff, to 1 decimal place?

Something to think about

A triangle has opposite 6 and adjacent 8, so the hypotenuse is 10. Find the angle three ways: with tan, with sin and with cos. What do you notice? Which one would you choose if you did not know the hypotenuse?

One way to think about it

tan⁻¹(6 ÷ 8) = tan⁻¹(0.75) = 36.9°, sin⁻¹(6 ÷ 10) = sin⁻¹(0.6) = 36.9° and cos⁻¹(8 ÷ 10) = cos⁻¹(0.8) = 36.9°. All three agree, as they must. If the hypotenuse is not known, use tan, because it needs only the opposite and adjacent sides.

Exit ticket

9. A triangle has an opposite side of 5 and an adjacent side of 9. Which expression gives the angle?