The three trigonometric ratios

1. The three trigonometric ratios O P P O S I T E H Y P O T E N U S Opposite Adjacent Opposite θ E Cos θ= Sin θ= Tan θ= Hypotenuse Hypotenuse Adjacent A D J A C E N T Remember: S O H C A H T O A S O H C A H T O A

2. Finding side lengths opposite 12 cm x sin θ= hypotenuse x sin 56°= 56° 12 If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example, Find x to 2 decimal places. We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use: x = 12 × sin 56° = 9.95 cm

3. Finding side lengths adjacent cos θ= hypotenuse x 5 A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground. What is the distance between the base of the ladder and the wall? We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use: 5 m 70° x cos 70°= x = 5 × cos 70° = 1.71 m (to 2 d.p.)

4. Finding side lengths

5. The inverse of sin sin–1 0.5 = sin 30° 0.5 sin–1 sin θ = 0.5, what is the value of θ? To work this out use the sin–1 key on the calculator. 30° sin–1 is the inverse of sin. It is sometimes called arcsin.

6. The inverse of cos cos–1 0.5 = cos 60° 0.5 cos–1 Cos θ = 0.5, what is the value of θ? To work this out use the cos–1 key on the calculator. 60° Cos–1 is the inverse of cos. It is sometimes called arccos.

7. The inverse of tan tan–1 1 = tan 45° 1 tan–1 tan θ = 1, what is the value of θ? To work this out use the tan–1 key on the calculator. 45° tan–1 is the inverse of tan. It is sometimes called arctan.

8. Finding angles 8 cm 5 cm θ tan θ= opposite adjacent 8 tan θ= 5 Find θto 2 decimal places. We are given the lengths of the sides opposite and adjacent to the angle, so we use: θ = tan–1 (8 ÷ 5) = 57.99° (to 2 d.p.)

9. Finding angles