Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 4 of 8

Sine, cosine and tangent without a calculator

A square cut along its diagonal and an equilateral triangle cut down the middle each hide ratios that stay the same at every size. This lesson finds them and puts them to work without a calculator.

Geometry and TrigonometryNo calculator

Warm-up

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

  1. From Year 10: simplify √50.
    Answer5√2, because 50 = 25 × 2.
  2. From Year 10: write 6/√3 with a whole-number denominator.
    Answer2√3, because 6/√3 = 6√3/3 = 2√3.
  3. From Year 9: a right-angled triangle has a hypotenuse of 10 cm and a 30° angle. Which ratio finds the side opposite the angle?
    AnswerSine: opposite = 10 × sin 30°.
Words used in this lesson
exact value
an answer written with roots or fractions and no rounding. For example: a root left unevaluated.
surd
a root that does not simplify to a whole number or a fraction. For example: the root of a prime number.
rationalize
rewrite so that the denominator is a whole number. For example: turning a fraction over a root into one over a whole number.
equilateral
a triangle with three equal sides. For example: the face of a regular tetrahedron.
isosceles
a triangle with two equal sides. For example: a triangle cut off a square by a diagonal.
altitude
the perpendicular distance from a vertex to the opposite side. For example: the height of a triangle.

A square cut in two

Take a square of side 1 and cut it along a diagonal. Each half is a right-angled triangle with two angles of 45°. The ratio sine of 45° uses one leg and the hypotenuse.

Predict first: is sin 45° a whole number, a simple fraction, or something with a square root in it? Write down what you think, and what the hypotenuse of a unit square's half is, then use the first figure to test it.

3
3√2 ≈ 4.24hypotenuse
0.7071sin 45° as leg ÷ hypotenuse

Legs 3, hypotenuse 3√2 ≈ 4.24. sin 45° = 3 ÷ 3√2 = 0.7071, which is 1 ÷ √2 = √2 ÷ 2.

Show me the steps
  1. Legs 3, hypotenuse 3√2 ≈ 4.24. sin 45° = 3 ÷ 3√2 = 1 ÷ √2 ≈ 0.7071, and tan 45° = 3 ÷ 3 = 1.
  2. Make the legs 1: hypotenuse √2 ≈ 1.41 and sin 45° = 1 ÷ √2 ≈ 0.7071. The ratio has not moved.
  3. Make the legs 8: hypotenuse 8√2 ≈ 11.31 and sin 45° = 8 ÷ 8√2 ≈ 0.7071. The size cancels, so one exact value serves every size.
  4. Rationalize: 1 ÷ √2 = (1 × √2) ÷ (√2 × √2) = √2 ÷ 2. It is the same number, 0.7071, with a whole-number denominator.

Two triangles give every exact value. Half a square has sides in the ratio 1 : 1 : √2, so sin 45° = cos 45° = 1/√2 = √2/2 and tan 45° = 1. Half an equilateral triangle has sides in the ratio 1 : √3 : 2, with the 30° angle opposite the shortest side.

30°: sin = 1/2, cos = √3/2, tan = 1/√3 = √3/3.
60°: sin = √3/2, cos = 1/2, tan = √3.
Rebuild them from the triangle whenever you need them: opposite and adjacent are named from the angle you are using.

8
4√3 ≈ 6.93altitude
√3 ≈ 1.73tan 60°

Side 8: the half has sides 4, 4√3 ≈ 6.93 and 8. sin 30° = 4 ÷ 8 = 0.5. tan 60° = 4√3 ÷ 4 = √3 ≈ 1.73.

Show me the steps
  1. Side 8: the half has sides 4, 4√3 ≈ 6.93 and 8. sin 30° = 4 ÷ 8 = 0.5 and cos 30° = 4√3 ÷ 8 = √3 ÷ 2 ≈ 0.87.
  2. Make the side 2: the half has sides 1, √3 and 2, the smallest version. sin 30° = 1 ÷ 2 and tan 60° = √3 ÷ 1.
  3. Make the side 12: sides 6, 6√3 and 12. tan 30° = 6 ÷ 6√3 = 1 ÷ √3 ≈ 0.58, the same as for every size.
  4. The 30° angle is opposite the shortest side, and the 60° angle is opposite the longer leg, so sin 30° = cos 60° and cos 30° = sin 60°.

The triangle decides, not memory. A common slip swaps the values for 30° and 60°. Check with the picture: the 30° angle is the small one, so it sits opposite the shortest side and its sine is the small value, 1/2. The 60° angle is opposite the longer leg.

Worked examples

1. A right-angled triangle has a 60° angle and a hypotenuse of 30 cm. Find the other two sides exactly.

  1. Adjacent to 60°: 30 × cos 60° = 30 × 1/2 = 15 cm.
  2. Opposite 60°: 30 × sin 60° = 30 × √3/2 = 15√3 cm.

2. Find the exact height of an equilateral triangle of side 22 cm.

  1. The altitude makes a right-angled triangle with hypotenuse 22 and angle 60° at the base.
  2. Height = 22 × sin 60° = 22 × √3/2 = 11√3 cm.

3. A rope of length 20 m runs from the top of a pole to the ground at an angle of 45°. How tall is the pole, exactly?

  1. The height is opposite the 45° angle: 20 × sin 45° = 20 × √2/2.
  2. That simplifies to 10√2 m.

Your go. The first step is given. A right-angled triangle has angles 30°, 60° and 90°, and its hypotenuse is 26 cm. Find the other two sides exactly.

  1. The side opposite the 30° angle is 26 × sin 30°.
  2. The side opposite the 60° angle is 26 × sin 60°. Leave it with a root.
Check the answer

26 × 1/2 = 13 cm and 26 × √3/2 = 13√3 cm. The longer leg is √3 ≈ 1.7 times the shorter, so the 13√3 is about 22.5 cm: bigger than 13 and smaller than 26.

Check before you go on

1. A right-angled triangle has angles 30°, 60° and 90°, and its shortest side is 5 cm. How long is its hypotenuse?

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. A right-angled triangle has a 30° angle and a hypotenuse of 14 cm. How long is the side opposite the 30° angle, in cm?

3. A ladder 8 m long leans against a wall at 45° to the ground. The height it reaches is ?√2 m. What number goes in front of the √2?

Level 2 · Building

4. In triangle ABC the angle at B is 90°, the angle at A is 60° and AB is 9 cm. BC is ?√3 cm. What number goes in front of the √3?

5. Without a calculator, work out tan 60° × cos 30°. Give your answer as a decimal, to 2 decimal places.

6. The sun is 30° above the horizon and a pole is 12 m tall. Its shadow is ?√3 m long. What number goes in front of the √3?

Level 3 · Stretch

7. An equilateral triangle has sides of 14 cm. Its height is ?√3 cm. What number goes in front of the √3?

8. A right-angled triangle has angles 30°, 60° and 90°, and its shortest side is 6 cm. Its area is ?√3 cm². What number goes in front of the √3?

Something to think about

Use the two triangles to find tan 45° and tan 60° exactly. Then decide, without a calculator, which is bigger: sin 60° or tan 45°?

One way to think about it

tan 45° = 1 and tan 60° = √3. For the comparison, sin 60° = √3 ÷ 2 and √3 is less than 2, so sin 60° is less than 1 = tan 45°. A sine never exceeds 1, and a tangent can.

Exit ticket

9. A right-angled triangle has angles 45°, 45° and 90°, and each of its two shorter sides is 12 cm. How long is the hypotenuse?