M4.3.6 Chapter 6

Exact values

On a calculator sin 45° shows 0.7071067812 and goes on. Is there a neat exact way to write it?

Your guess

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

Your guess

A No. The decimal never ends, so it cannot be written exactly.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Yes. It is exactly 0.7071, the calculator is only adding extra digits.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Yes. It is 1 ÷ √2, and half a square shows why.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Recall sin, cos and tan of 30°, 45° and 60°
Work out those values from two special triangles

A calculator gives sin 60° as 0.8660254... The digits never end and never repeat, so no decimal is exact. But for 30°, 45° and 60° we can write the exact value, as a fraction or with a square root. We get them from two triangles that take a minute to draw.

First triangle. Draw a square with sides of 1 cm. Cut it along a diagonal. You get two equal right angled triangles.

The diagonal is √2 cm because 1² + 1² = 2.

The diagonal cuts each right angle in half. So each triangle has angles 45°, 45° and 90°. Its sides are 1 cm, 1 cm and √2 cm.

Worked example

Use half a square with sides 1 cm to find sin 45°, cos 45° and tan 45° exactly.

Draw a square with sides 1 cm and cut it along a diagonal.
Each half has a right angle, two angles of 45° and two sides of 1 cm.
The hypotenuse by Pythagoras: c² = 1² + 1² = 2, so c = √2 cm.
For a 45° angle the opposite side is 1 and the hypotenuse is √2, so sin 45° = 1 ÷ √2.
The adjacent side is 1, so cos 45° = 1 ÷ √2 and tan 45° = 1 ÷ 1 = 1.

Answer: sin 45° = 1 ÷ √2, cos 45° = 1 ÷ √2 and tan 45° = 1.

You may meet √2 ÷ 2 instead of 1 ÷ √2. They are the same number. Multiply the top and the bottom of 1 ÷ √2 by √2, and you get √2 ÷ 2. Both are 0.7071... Many books prefer √2 ÷ 2 because it has no root on the bottom.

Second triangle. Draw an equilateral triangle with sides 2 cm. Cut it down the middle from the top corner to the middle of the base.

The cut line is √3 cm because 2² − 1² = 3.

The cut makes two right angled triangles. The top angle of 60° is split into two 30° angles. Each triangle has angles 30°, 60° and 90°, and sides 1 cm, √3 cm and 2 cm.

Worked example

Use half an equilateral triangle with sides 2 cm to find sin 30°, cos 30° and tan 30° exactly.

Draw an equilateral triangle with sides 2 cm and cut it in half down the middle.
Each half has a right angle and angles of 30° and 60°. Its hypotenuse is 2 cm and its short side is 1 cm.
The third side by Pythagoras: h² = 2² − 1² = 3, so h = √3 cm.
For the 30° angle the opposite side is 1 and the hypotenuse is 2, so sin 30° = 1/2.
The adjacent side is √3, so cos 30° = √3 ÷ 2 and tan 30° = 1 ÷ √3.

Answer: sin 30° = 1/2, cos 30° = √3 ÷ 2 and tan 30° = 1 ÷ √3.

The same triangle gives the 60° values. The 60° angle is at the other end, so opposite and adjacent swap. Then sin 60° = √3 ÷ 2, cos 60° = 1/2 and tan 60° = √3 ÷ 1 = √3.

Learn this table, or learn how to draw the two triangles and read it off.

In the table a slash is a division. So 1/√3 means 1 ÷ √3, and √3/2 means √3 ÷ 2. The two ways of writing say the same thing.

Exact values keep an answer exact. A wire 10 m long holds a pole upright. It makes an angle of 60° with the ground. How high up the pole is it fixed? Leave the answer with a root first, then round it.

Worked example

A wire 10 m long holds a pole upright and makes an angle of 60° with the ground. How high up the pole is it fixed? Give it exactly, then to 1 decimal place.

The height is opposite the 60° angle. The wire is the hypotenuse. We have O and H, so use sin.
sin 60° = √3 ÷ 2, so h = 10 × √3 ÷ 2.
Work out 10 ÷ 2 = 5.
So h = 5 × √3 = 5√3 m, exactly.
On the calculator, 5 × 1.7321 = 8.6605.
To 1 decimal place, h = 8.7 m.

Answer: The wire is fixed exactly 5√3 m up the pole, which is 8.7 m to 1 decimal place.

See it three ways

With things

Fold a square sheet of paper corner to corner. The fold is longer than a side, and the two halves are the same.

As a picture

In symbols

sin 45° = 1 ÷ √2, which is about 0.7071.

Try it

Fold a square

You need: a square sheet of paper, a ruler.

Cut a square with sides of 10 cm. Fold it corner to corner.
Measure the fold, which is the hypotenuse of each half.
Divide a side, 10 cm, by the fold.

What you will see: The fold is about 14.1 cm. The division 10 ÷ 14.1 is about 0.71. This is sin 45° = 1 ÷ √2, about 0.7071, as in the table.

If you cannot try it, this is what happens.

A wrong idea many people have

cos 30° is 1/2, because sin 30° is 1/2 and the two go together.

Why it feels right: The numbers 1/2 and √3 ÷ 2 appear together in the table, and the two are often swapped.

The test that breaks it: Draw the 30° triangle. The side next to the 30° angle is the long leg, √3 cm, not the short one. So cos 30° = √3 ÷ 2, about 0.87. The calculator agrees.

What is true: sin 30° = 1/2 and cos 30° = √3 ÷ 2. Cos 60° is the one that equals 1/2. The sine and the cosine of 30° and 60° swap.

A wrong idea many people have

1 ÷ √2 and √2 ÷ 2 are two different numbers.

Why it feels right: They look different, and one has a root on the bottom.

The test that breaks it: Multiply the top and the bottom of 1 ÷ √2 by √2. The top becomes 1 × √2 = √2. The bottom becomes √2 × √2 = 2. So 1 ÷ √2 = √2 ÷ 2.

What is true: They are the same number, 0.7071... Writing a root off the bottom is a matter of habit, not of value.

Key words

exact value: a value written with a fraction or a root, not rounded to decimals.

Back to your guess

The answer is C: Yes. It is 1 ÷ √2, and half a square shows why.

It is exactly 1 ÷ √2. A decimal that never ends can still be written exactly with a root. Half a square gives sin 45°, and half an equilateral triangle gives the values for 30° and 60°. In this chapter you will see where each comes from.

Keep it

Copy this into your exercise book.

Half a square gives 45°: sides 1, 1, √2. Half an equilateral triangle gives 30° and 60°: sides 1, √3, 2. Read sin, cos and tan from these.

Draw: The two triangles with their sides and angles, and a small table of sin, cos and tan for 30°, 45° and 60°.