1
Exact ratios of 30°, 45° and 60° from the equilateral triangle and the square
Mathematics Prep 3 Term 1, Unit 3, Lesson 5
GoalFind sin, cos and tan of 30°, 45° and 60° exactly, and use them in expressions and sides.
New words
equilateral triangle · a triangle with three equal sides and three 60° anglesaltitude · the perpendicular from a vertex to the opposite sidediagonal of a square · a square of side 1 has diagonal √2exact value · an answer kept as a fraction or with √, not rounded
Explanation
Take an equilateral triangle of side 2 and draw one altitude. It cuts the triangle into two right triangles with angles 30°, 60° and 90°. The short leg is 1 (half the side), the hypotenuse is 2, and the altitude is √(4 − 1) = √3. Read the ratios: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and for 60° they swap: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. Now a square of side 1: the diagonal is √2 and makes 45° with a side. So sin 45° = cos 45° = 1/√2 = √2/2 and tan 45° = 1. These nine values are worth remembering, since they appear without a calculator.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Worked examples
In the half-equilateral triangle in the picture (30°, 60°, 90°) with hypotenuse 10 m: the side opposite 30° is 10 × sin 30° = 5 m, and the side adjacent to 30° is 10 × cos 30° = 5√3 m.
Equilateral triangle of side 8 m. Its altitude makes 60° with the base, so the altitude = 8 × sin 60° = 8 × √3/2 = 4√3 m.
Square of side 5 cm. The diagonal makes 45° with a side, so the diagonal = 5 ÷ cos 45° = 5 ÷ (√2/2) = 5√2 cm.
Evaluate sin 60° cos 30° − cos 60° sin 30° = (√3/2)(√3/2) − (1/2)(1/2) = 3/4 − 1/4 = 1/2.
Class activity
Fold and measure. Cut a square of paper and fold it along the diagonal. Measure side and diagonal with a ruler and divide: diagonal ÷ side should be close to 1.414 (that is √2).
Practice
1
Write sin 30°, cos 60° and tan 45° exactly.
1/2, 1/2 and 1.
2
Evaluate sin²30° + cos²60°.
(1/2)² + (1/2)² = 1/4 + 1/4 = 1/2.
3
Evaluate tan 60° × tan 30°.
√3 × 1/√3 = 1.
4
A 30°-60°-90° triangle has hypotenuse 30 m. Find the short leg.
30 × sin 30° = 15 m.
5
Show that sin 45° × cos 45° = 1/2.
(√2/2)(√2/2) = 2/4 = 1/2.