Exact trigonometric values

The Maths Habit

Give each angle to 1 decimal place.

  1. Find angle x.

    x = 24.6°

  2. Find angle x.

    x = 50.0°

  3. Find angle x.

    x = 61.9°

Is 0.33 exactly one third? Is 0.333333? How could you write one third exactly?

Both are rounded: only 13 is exact

Exact value
A value written with fractions or surds, not a rounded decimal.

Cut an equilateral triangle with sides of 2 in half. What are the angles and sides of each half?

Angles 30°, 60°, 90°

The base halves to 1

Find the missing side of the half. Leave it as a surd.

? = √(2² − 1²)

? = √3

A square with sides of 1. Cut it along a diagonal. Find the angles and the diagonal.

Angles 45°, 45°, 90°

? = √(1² + 1²) = √2

Half an equilateral triangle

Half a square

Use the triangle to write the exact value of sin 30°.

Use the triangle to write the exact value of cos 30°.

tan 30° =oppositeadjacent=1√3
tan 30° =1√3
cos 45° =adjacenthypotenuse=1√2
cos 45° =1√2
tan 30° =oppositeadjacent=1√3
tan 30° =1√3
sin 45° =oppositehypotenuse=1√2
sin 45° =1√2

Exact trigonometric values worksheet, page 1

Exact trigonometric values worksheet, page 2

Answer sheet

  1. Without a calculator, work out sin 30° + cos 60°.

    1

  2. Show that tan 30° × tan 60° = 1.

    1√3 × √3 = 1

  3. A right-angled triangle has a hypotenuse of 14 cm and an angle of 60°. Find the exact length of the side opposite the 60° angle.

    14 × √32 = 7 √3 cm

  4. A right-angled triangle has an angle of 60°. The side next to the 60° angle, not the hypotenuse, is 7 cm. Find the exact length of the side opposite the 60° angle.

    7 × √3 = 7 √3 cm

  1. Write the exact value of cos 30°.

    √32

  2. Write the exact value of cos 45°.

    1√2 or √22

  3. Find the exact length of x.

    x = 5√3 cm
    5 cmx60°
Exit tickets to print