Trigonometry-5 Examples and Problems

Trigonometry Working with Greek letters to show angles in right angled triangles. Exercises

What you have Learned  In any right angled triangle we can find the ratio of sides.  The ratios are: O  A H O A Tan  = O H Sin  = A H Cos  = Some Old Hens Cackle And Howl Till Old Age

 In triangle ABC angle A = 25  and AB = 7,8 units. What is the length of BC?  Let BC = x Example-1 25  Ask yourself – we are working with angle A: A B C x 7,8 Do we know the opposite side to 25  ? Do we know the adjacent side to 25  ? Do we know the hypotenuse? Yes - x No Yes – 7,8 Work only with what you know or want to find. We don’t know the adjacent side, so leave it out.

 Which trig ratio does not use the adjacent side? Example-1 25  A B C x 7,8 O A Tan  = O H Sin  = A H Cos  = x x This ratio does not use the adjacent side. This is the ratio we must use.

 Start with the sine ratio and put in the values. Example-1 25  A B C x 7,8 O H Sin  = x 7,8 Sin 25 1 = To get x on its own multiply both sides by 7,8. 7,8 1 1 Now simplify. xSin 25=7,8 Find sin25  on calculator. x0,4226=7,8 x = 3,296

 In triangle PQR angle P = 37  and PQ = 11,3 units. What is the length of PR?  Let PR = x Example-2 37  Ask yourself – we are working with angle P: P Q R x 11,3 Do we know the opposite side to 37  ? Do we know the adjacent side to 37  ? Do we know the hypotenuse? No Yes - x Yes – 11,3 Work only with what you know or want to find. We don’t know the opposite side, so leave it out.

 Which trig ratio does not use the opposite side? Example-2 37  P Q R 11,3 O A Tan  = O H Sin  = A H Cos  = x x This ratio does not use the opposite side. This is the ratio we must use. x

 Start with the cosine ratio and put in the values. Example-2 37  P Q R 11,3 A H cos  = x 11,3 cos 37 1 = To get x on its own multiply both sides by 11,3. 11,3 1 1 Now simplify. xcos 37=11,3 Find cos37  on calculator. x0,7986=11,3 x = 9,024 x

 In triangle PQR angle P = 52  and QR = 6,7 units. What is the length of PQ?  Let PQ = x Example-3 52  Ask yourself – we are working with angle P: P Q R x 6,7 Do we know the opposite side to 52  ? Do we know the adjacent side to 52  ? Do we know the hypotenuse? Yes–6,7 No Yes – x Work only with what you know or want to find. We don’t know the adjacent side, so leave it out.

 Which trig ratio does not use the adjacent side? Example-3 52  P Q R O A Tan  = O H Sin  = A H Cos  = x x This ratio does not use the adjacent side. This is the ratio we must use. x 6,7

 Start with the sine ratio and put in the values. Example-3 52  P Q R O H sin  = 6,7 x sin 52 1 = To get x on its own on top multiply both sides by x. x 1 x 1 Cancel the x’s. 6,7sin 52 = x Divide both sides by sin52. x = 8,502 x 6,7 Simplify and find sin52. sin 52 x = 6,7 = 0,788 6,7

 In triangle PQR angle P = 52  and QR = 6,7 units. What is the length of PR?  Let PR = x Example-4 52  Ask yourself – we are working with angle P: P Q R x 6,7 Do we know the opposite side to 52  ? Do we know the adjacent side to 52  ? Do we know the hypotenuse? Yes–6,7 Yes-x No Work only with what you know or want to find. We don’t know the hypotenuse, so leave it out.

 Which trig ratio does not use the hypotenuse? Example-4 52  P Q R O A Tan  = O H Sin  = A H Cos  = x x This ratio does not use the hypotenuse. This is the ratio we must use. 6,7 x

 Start with the tan ratio and put in the values. Example-4 52  P Q R O A tan  = 6,7 x tan 52 1 = To get x on its own on top multiply both sides by x. x 1 x 1 Cancel the x’s. 6,7tan 52 = x Divide both sides by tan52. x = 5,238 6,7 Simplify and find tan52. tan 52 x = 6,7 = 1,279 6,7 x

 You have to find the height of a tree.  From where you stand, 76 metres from the tree, the angle to the top of the tree is 32 . A Typical Problem Always draw a diagram. 32  Ground Tree Line of sight You don’t have to draw a work of art, just a simple sketch.

32   The angle is 32  and we are 76 metres from the tree. How high is the tree?  Let tree height = x Problem-1 Ask yourself – we are working with angle 32  : x 76 Do we know the opposite side? Do we know the adjacent side Do we know the hypotenuse? Yes - x Yes - 76 No Work only with what you know or want to find. We don’t know the hypotenuse, so leave it out.

 Which trig ratio does not use the hypotenuse? Problem-1 O A Tan 32 = O H Sin 32 = A H Cos 32 = x x This ratio does not use the adjacent side. This is the ratio we must use. x 32  76

 Start with the tan ratio and put in the values. Problem-1 O A Tan 32 = x 76 Tan 32 1 = To get x on its own multiply both sides by Now simplify. xTan 32=76 Find tan 32  on calculator. x0,6248=76 x = 4,873 x 32  76

 A surveyor has to find the distance from point A to point B on the other side of a lake. The distance AC is measured to be 275 m, and angle A is 40 . A Typical Problem If you are given a diagram, then don’t waste time drawing another one. 40  275 m A B C

 In triangle ABC angle A is 40  and AC = 275 m. What is the length of AB?  Let AB = x Example-4 Ask yourself – we are working with angle A: Do we know the opposite side to 40  ? Do we know the adjacent side to 40  ? Do we know the hypotenuse? No Yes - x Yes Work only with what you know or want to find. We don’t know the opposite side, so leave it out. 40  275 m A B C x

 Which trig ratio does not use the opposite side? Example-4 O A Tan  = O H Sin  = A H Cos  = x x This ratio does not use the opposite side. This is the ratio we must use. 40  275 m A B C x

 Start with the cos ratio and put in the values. Example-4 A H cos  = x 275 cos 40 1 = To get x on its own multiply both sides by Cancel the 275’s. xcos 40 = 275 Find cos40 and multiply. x = 210,7m 40  275 m A B C x x0.766 = 275

Well Done!