Lesson 12 of 14
The big picture
Show on a big screenOne triangle, one angle, six ratios. Every value here comes from a right triangle and one angle.
| Idea | Remember | Example |
|---|---|---|
| Naming the sides | Opposite faces the angle. Adjacent touches it. | At angle A: opposite, adjacent, hypotenuse |
| Three ratios | In a right triangle, for acute A: sin A = opposite ÷ hypotenuse. cos A = adjacent ÷ hypotenuse. tan A = opposite ÷ adjacent. | 3-4-5 gives 4/5, 3/5, 4/3 |
| Size does not matter | Angle A alone decides sin A, cos A and tan A. | 3-4-5 and 6-8-10 both give sin A = 4/5 |
| The flips | cosec A = 1 ÷ sin A. sec A = 1 ÷ cos A. cot A = 1 ÷ tan A. | sin A × cosec A = 1 |
| 0° and 90° | sin 0° = 0, cos 0° = 1, tan 0° = 0. sin 90° = 1, cos 90° = 0. tan 90° is not defined. | tan 90° has no value |
| 45° | sin 45° = 1/√2. cos 45° = 1/√2. tan 45° = 1. | Half a square: legs 1, 1 and √2 |
| 30° and 60° | sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. | Half an equilateral: sides 1, √3, 2 |
| tan from sin and cos | tan A = sin A ÷ cos A. cot A = cos A ÷ sin A. | 5/13 ÷ 12/13 = 5/12 |
| The identity | sin²A + cos²A = 1 for every acute angle A. | 16/25 + 9/25 = 1 |
| Two more | A is an acute angle. 1 + tan²A = sec²A and 1 + cot²A = cosec²A. | Divide the identity by cos²A or sin²A |
Chapter 8 is Introduction to Trigonometry. Open it and read every exercise as a triangle to name.