Skip to content
Cambridge IGCSE0607

Exact trigonometric values

International Mathematics 0607 Chapter Notes

What this chapter covers

Exact trigonometric values
ShareWhatsAppPost
Exact trigonometric values notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Exact trigonometric values notes as text: skim, search, and jump between subtopics.

~4 min read

In this chapter

  1. 1Sine and cosine of 0° to 90°
  2. 2Tangent of 0° to 60°
  3. 3Getting the values back from two triangles

1. Sine and cosine of 0° to 90°

Know the exact values of sin x and cos x for x = 0°, 30°, 45°, 60° and 90°.

An exact value is written with fractions and surds, never as a rounded decimal. For example, (√3)/2 is exact; 0.866 is not.

The cosine values are the sine values in reverse order. So sin 30° = ½ but cos 30° is (√3)/2.

beginarrayc|ccccc x & 0^(°) & 30^(°) & 45^(°) & 60^(°) & 90^(°) hline sin x & 0 & 1/2 & (√2)/2 & (√3)/2 & 1 cos x & 1 & (√3)/2 & (√2)/2 & 1/2 & 0 endarray

Examiner insight

“Often answers were given as decimals rather than in surd form, indicating that candidates need to know what the ‘exact value’ means.”

Worked example 11 mark

Write down the exact value of cos 30°.

  1. 1

    cos 30^(°) = (√3)/2 [1]

2. Tangent of 0° to 60°

Know the exact values of tan x for x = 0°, 30°, 45° and 60°. The list stops at 60°.

The values rise from 0 to √3, with tan 45° = 1 in the middle. So tan 30° is (√3)/3, which is less than 1, and tan 60° is √3, which is more than 1.

In a right-angled triangle the unknown side is often underneath the fraction. Rearrange one line at a time.

beginarrayc|cccc x & 0^(°) & 30^(°) & 45^(°) & 60^(°) hline tan x & 0 & (√3)/3 & 1 & √3 endarray

Examiner insight

“The majority of candidates realised that a tan ratio was required, but some of these candidates could not recall the exact value of tan 60.” “There were a significant number of careless slips in rearranging a correct equation.”

Worked example 13 marks

Do not use a calculator.

A right-angled triangle has an angle of 60°. The side opposite this angle is 9 cm. The side next to this angle is x cm. Work out the exact value of x.

  1. 1

    tan 60^(°) = 9/x [1]

  2. 2

    tan 60^(°) = √3, so x = 9/(√3) [1]

  3. 3

    x = (9√3)/3 = 3√3 [1]

3. Getting the values back from two triangles

If you forget a value, draw a triangle and use Pythagoras' theorem.

Half an equilateral triangle of side 2 has sides 1, √3 and 2, with angles 30°, 60° and 90°. A right-angled isosceles triangle with two sides of 1 has a hypotenuse of √2, with angles 45°, 45° and 90°.

Read each ratio from the triangle: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.

sin 45^(°) = 1/(√2) = (√2)/2

tan 30^(°) = 1/(√3) = (√3)/3

Examiner insight

“Some candidates demonstrated their knowledge by showing all the exact sine, cosine and tangent values for angles of 30°, 45° and 60°, usually in a table, before selecting the correct answer.” “Other candidates obtained the correct answer by drawing out a suitable triangle and using Pythagoras’ theorem to show that sin30° = 0.5.”

Worked example 14 marks

Do not use a calculator.

In triangle ABC, angle B = 90°, angle A = 30° and AC = 10 cm. Work out the exact lengths of BC and AB.

  1. 1

    BC is opposite angle A: BC = 10 sin 30^(°) [1]

  2. 2

    BC = 10 × 1/2 = 5 cm [1]

  3. 3

    AB is adjacent to angle A: AB = 10 cos 30^(°) [1]

  4. 4

    AB = 10 × (√3)/2 = 5√3 cm [1]

Recap

  • Exact values use fractions and surds, not rounded decimals.
  • sin x for 0°, 30°, 45°, 60°, 90°: 0, 1/2, (√2)/2, (√3)/2, 1.
  • cos x for 0°, 30°, 45°, 60°, 90°: 1, (√3)/2, (√2)/2, 1/2, 0.
  • tan x for 0°, 30°, 45°, 60°: 0, (√3)/3, 1, √3.
  • Forgotten? Draw half an equilateral triangle with sides 1, √3, 2 or a right-angled isosceles triangle with sides 1, 1, √2.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the exact value of sin 30°.1 mark
  2. Do not use a calculator. Write down the exact value of cos 60°.1 mark
  3. Do not use a calculator. Write down the exact value of tan 45°.1 mark
  4. Do not use a calculator. Work out sin 90^(°) + cos 90^(°) + cos 0^(°).2 marks
  5. Do not use a calculator. Work out the exact value of sin 45^(°) × cos 45^(°).2 marks
  6. Do not use a calculator. A right-angled triangle has an angle of 30°. The side next to this angle is 6 cm. Work out the exact length of the side opposite this angle.3 marks
  7. Do not use a calculator. A right-angled triangle has a hypotenuse of 14 cm. One of the other sides is 7 cm. Find the angle opposite the 7 cm side and the exact length of the third side.3 marks
  8. Do not use a calculator. Work out the exact value of tan 60^(°) - tan 30^(°).3 marks
  9. Do not use a calculator. Write these five values in order of size, smallest first. Give the exact value of each one in your working. sin 60^(°) tan 30^(°) cos 60^(°) tan 60^(°) cos 45^(°)4 marks
  10. Do not use a calculator. A vertical mast TF stands on horizontal ground with its foot at F. The points A, B and F are in a straight line on the ground, with B between A and F. AB = 20 m, angle TAF = 30° and angle TBF = 60°. (a) Work out the exact height of the mast. (b) Work out the length of BT.7 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations