Exact values of sin, cos and tan of 0°, 30°, 45°, 60° and 90° from two special triangles, then Pythagoras and trigonometry in cuboids and pyramids (Higher).
What you will learn
- Know sin, cos and tan of 0°, 30°, 45°, 60° and 90° exactly
- Use exact values without a calculator
- Give a length as a surd in its simplest form (Higher)
- Find lengths and angles in cuboids and pyramids (Higher)
Two triangles give every value
Cut a square with sides 1 along its diagonal: the hypotenuse is √2 and both angles are 45°.
Cut an equilateral triangle with sides 2 in half: the sides are 1, √3 and 2, with angles 30° and 60°.
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | no value |
√2/2 is the same as 1/√2, and √3/3 is the same as 1/√3. All tiers need this table.
Read the table both ways. If tan x = 1, then x = 45°. If 2 sin x = 1, first divide by 2: sin x = 1/2, so x = 30°. If √3 tan x = 3, then tan x = 3/√3 = √3, so x = 60°.
To combine exact values, replace each one and simplify, using √3 × √3 = 3 and √2 × √2 = 2: 4 × sin 30° × cos 60° = 4 × 1/2 × 1/2 = 1, and tan 60° × tan 30° = √3 × 1/√3 = 1.
For an exact side, use the exact value in place of the decimal: cos 30° = x/10 gives x = 10 × √3/2 = 5√3. When a √ ends up on the bottom, multiply top and bottom by it: 6/√3 = 6√3/3 = 2√3.
Pythagoras can give an exact surd answer too. Write the square root in its simplest form by taking out a square factor: x² = 4² + 8² = 80, so x = √80 = √(16 × 5) = 4√5.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- Two triangles give every value
- Pythagoras and trigonometry in 3D (Higher)
- Try it: Read the exact values off the circle
7 practice questions, marked as you go; about 25 minutes in all.
In this unit: Pythagoras and trigonometry
Pythagoras' theorem, sin, cos and tan in right-angled triangles, angles of elevation, depression and bearings, exact values, 3D problems, and the sine and cosine rules.