Trigonometry

1. Trigonometry

2. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle Both the triangles below are similar because: The angles are the same but the sides are different

3. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle Both the triangles below are similar because: The angles are the same but the sides are different

4. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle Both the triangles below are similar because: The angles are the same but the sides are different

5. For both triangles If we measure the height and the base: 5 cm 8 cm 10 cm 16 cm

6. This angle is in fact 320 So as long as the value of then this angle will always be 320

7. This is the idea behind trigonometry If we know 2 sides then we can find the angles in the triangle How do we know the angle is 320 ? We can use our calculator which has been programmed to work out the angle.

8. We don`t have to know the height and the base it can be any 2 sides Depending on which 2 sides are known then we use a different button on the calculator Names are given to the 3 sides which all refer to the angle we are trying to find

9. The names are: Opposite, Adjacent and Hypotenuse Opposite means on the other side from the angle we need. Adjacent means next to the angle we need. Hypotenuse means the side opposite the right angle Hypotenuse Opposite X Adjacent

10. Identify the names of the sides of these right angled-triangles given angle k opposite b opposite a b hypotenuse c c adjacent hypotenuse a k a opposite adjacent k c adjacent b k hypotenuse c opposite hypotenuse b k a adjacent

11. X X X X X X X X X X Opposite, Adjacent and Hypotenuse In each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”. x

12. Using the Opposite (O), Adjacent (A) and Hypotenuse (H) to work out the missing angle The calculator has 3 buttons which are used to find the missing angle: Sin – short for Sine Cos – short for Cosine Tan – short for Tangent

13. SOH CAH TOA • Memory Aid • Some Old Horses Sin Opposite Hypotenuse • Can Always Hear Cos Adjacent Hypotenuse • Their Owners Approaching Tan Opposite Adjacent • Or invent one of your own Deciding which button to use depends on which sides are given

14. O A O A T S H C H SOH CAH TOA • Divide it up into three groups • Place each group of three in a triangle starting in the bottom left of each triangle

15. O A O A T S H C H SOH Trigonometric Ratios CAH TOA

16. 25 cm x Cos (x) = Adjacent Hypotenuse 10 cm SOH CAH TOA Example 1 What have we got and need to find? We need an angle – x. We have the Hypotenuse and Adjacent side. Looking at the phrase, we can use C A H Hypotenuse Adjacent

17. 25cm x Cos (x) = Adjacent Hypotenuse 10 cm Replace A and H by 10 and 25 Hypotenuse Cos (x) = = 0.4 We now need to convert this to an angle in degrees using the Cos-1 button!!! Adjacent x = Cos –1(0.4) = 66.42o We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

18. Opposite Adjacent Tan (x) = SOH CAH TOA Example 2 What have we got and need to find? We need an angle – x. We have the Opposite and Adjacent side. Looking at the phrase, we can use TOA 15 cm Opposite x 20 cm Adjacent

19. We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons. Tan (x) = Opposite Adjacent Replace O and A by 15 and 20 = 0.75 Tan (x) = We now need to convert this to an angle in degrees using the Tan-1 button!!! 15 cm Opposite x x = Tan –1(0.75) = 36.67o 20 cm Adjacent

20. Sin (x) = Opposite Hypotenuse SOH CAH TOA Example 3 What have we got and need to find? We need an angle – x. We have the Hypotenuse and Opposite side. Looking at the phrase, we can use S O H Hypotenuse 12 cm 8 cm Opposite x

21. Sin (x) = Opposite Hypotenuse We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons. Replace O and H by 8 and 12 Sin (x) = = 0.666 Hypotenuse We now need to convert this to an angle in degrees using the Sin-1 button!!! 12 cm 8 cm Opposite x x = Sin –1(0.666) = 41.81o

22. Using Trigonometry to Find a Missing Side

23. O A O A T S H C H SOH Trigonometric Ratios CAH TOA

24. O S H SOH Trigonometric Ratios The triangle can also be used to find either the opposite side or the hypotenuse

25. A C H CAH Trigonometric Ratios The triangle can also be used to find either the adjacent side or the hypotenuse

26. O A T TOA Trigonometric Ratios The triangle can also be used to find either the adjacent side or the opposite

27. 60o H O Opposite Sin (angle) Hypotenuse = 3 m S H What have we got and need to find? We need the Hypotenuse H Example 1 SOH CAH TOA We have an angle and the Opposite O Hypotenuse Looking at the phrase we can use S O H Opposite

28. 3 Sin (60o) H = Use the Sin button on your calculator to find this value H = Opposite Sin (angle) Hypotenuse = Replace O by 3 and (angle) by 60o 60o H Hypotenuse H = 3.46410….. 3 m H = 3.46 m to 2 d.p. Opposite

29. O T A What have we got and need to find? We need the Adjacent A 40o Example 2 SOH CAH TOA We have an angle and the Opposite O A Adjacent Looking at the phrase we can use T O A 3 m Opposite

30. 3 Tan (40o) H = Use the Tan button on your calculator to find this value H = Replace O by 3 and (angle) by 40o 40o A Adjacent H = 3.575….. 3 m H = 3.58 m to 2 d.p. Opposite

31. A C H What have we got and need to find? We need the Adjacent A Example 3 SOH CAH TOA We have an angle and the Hypotenuse H 8 Hypotenuse Looking at the phrase we can use C A H 70o A Adjacent

32. Use the Cos button on your calculator to find this value Replace H by 8 and (angle) by 70o A = cos70 x 8 8 H = 0.342 x 8 Hypotenuse H = 2.736….. 70o A H = 2.74 m to 2 d.p. Adjacent