Trigonometry Part II Math 416 Game Plan

Trigonometry Part II Math 416

Game Plan § Area of Triangles § Traditional § Sine § Hero’s § Sine Law § Two examples § Word Problem

Traditional Area of Triangles § We are now going to be working with other triangles besides right angle triangles § The 1 st concept we will look at is area § Area of triangle = half the base x height § A = ½ b h or A = bh 2

Using Sin for Area of Triangle § We note the base is a side of the triangle § The height must be at 90° or perpendicular to that base (not up and down) Note b =8 § Eg 9 8 h =9 A = ½ bh A = ½ (8)(9) A = 36

Sin § Now let us try and get a different perspective. § Consider We are given two sides and the contained angle. From trig Sin θ = h p p h = p Sin θ θ Create formula for A q From formula A = ½ bh A = ½ qp Sinθ

Another Example Find Area 12 A = ½ (18)(12) Sin 47° 18 A = 78. 99

Using Hero’s to find Area of Triangle § Now a totally different approach was found by Hero or Heron § His approach is based on perimeter of a triangle

Be My Hero and Find the Area § Consider P = a + b + c (perimeter) p = a + b + c / 2 or p = P / 2 (semi perimeter) b A = p (p-a) (p-b) (p-c) a c Hence, by knowing the sides of a triangle, you can find the area

Be My Hero and Find the Area P = 9 + 11 + 8 = 28 p = 14 A = p (p-a) (p-b) (p-c) § Eg 9 11 A = 14(14 -9)(14 -11)(14 -8) A = 14 (5) (3) (6) 8 A = 1260 A = 35. 5

Be My Hero and Find the Area P = 42 + 43 + 47 p = 66 A = p (p-a) (p-b) (p-c) § Eg 42 43 A = 66(24)(23)(19) 47 A = 692208 A = 831. 99

Be My Hero and Find the Area P=9+7+3 p = 9. 5 A = p (p-a) (p-b) (p-c) § Eg 9 7 3 A = 9. 5(0. 5)(2. 5)(6. 5) A = 77. 19 A = 8. 79

Sin Law With respect to Angle A § Consider Sin A = h/b h = b sin A C With respect to Angle B Sin B = h/a b h = a sin B a h Thus, a. Sin. B = b. Sin. A Divide both sides by ab B A Sin B = Sin A b a

Sin Law § Now we can do this again using Angle C § What we get is the Sin Law for side lengths § a = b = c Sin A Sin B Sin C Sin Law for angles § Sin A = Sin B = Sin C a b c Notice when getting angles Sin on TOP (think a on top). .

Notes § Each expression is actually 3 formulae § You do not need the whole thing § Always look for the Side – Angle - Combo

1 st Example § Eg 8 57° β Complete the Triangle Let’s get angle or θ 1 st 15 θ Sin θ = Sin 57 8 15 x θ = 27°… now Beta β = 180 – 57 – 27 = 96°… now x x = 15 x = 17. 79 Sin 96 Sin 57

2 nd Example § Eg y 75° y = Sin 43 θ x 43° 18 18 Sin 62 Complete the Triangle Let’s get θ 1 st θ = 180 – 75 = 62… x? x = 18 Sin 75 Sin 62 x = 19. 69… now y? y = 13. 90

Word Problem § A surveyor creates the following map x = 200 Sin 73 Sin 63 Billy’s House x = 214. 66 y = 200 Sin 44 Sin 63 200 m y = 155. 93 Dist = 63° 73° 214. 66+155. 93+200 School Post Office = 570. 59 m § What is the shortest distance if Billy goes from home to school, to the post office and home?