[FONT=Verdana, Arial, Helvetica][SIZE=-1] Before now, we have been able to use the trigonometric functions to solve right triangles. This doesn't do much good because there are many times triangles will not be right triangles. The Law of Sines is one way that we can solve oblique triangles. It is listed below. [/SIZE][/FONT]
a b c ----- = ----- = ----- sin A sin B sin C [FONT=Verdana, Arial, Helvetica][SIZE=-1] When you know any two angles and any side of a triangle, you can use the law of sines to solve the triangle. Example: [/SIZE][/FONT]
1. Problem: In triangle ABC, a = 4.56, A = 43[SUP]o[/SUP], and C = 57[SUP]o[/SUP]. Solve the triangle. Solution: First, sketch the triangle and include any given information. Here's our idea of a sketch:
Find
angle B.
Angle B = 180[SUP]o[/SUP] - (43[SUP]o[/SUP] + 57[SUP]o[/SUP]) =
80[SUP]o[/SUP] Now, we use the
law of sines to find the other sides lengths.
c a ----- = ----- sin C sin A a(sin C) c = -------- sin A Plug in any known information.
4.56(sin 57[SUP]0[/SUP]) c = ------------- sin 43[SUP]o[/SUP] Use a calculator to find the sines.
4.56(.8387) c = ----------- .6820 c = 5.61 Now solve for
b.
b a ----- = ----- sin B sin A a(sin B) b = -------- sin A Plug in any known information.
4.56(sin 80[SUP]o[/SUP]) b = ------------- .6820 Use a calculator to find the sines.
4.56(.984 b = ----------- .6820 b = 6.58 [FONT=Verdana, Arial, Helvetica][SIZE=-1] There is another case when you can use the law of sines. When you know two sides and an angle opposite one of the sides, the law of sines can be used. However, with this case, you have to be aware that there might not be a solution, or there may be two! One solution is also possible. Example: [/SIZE][/FONT]
2. Problem: In triangle ABC, a = 15, b = 25, and angle A = 47[SUP]o[/SUP]. Solve the triangle. Solution: Start out by looking for the measure of
angle B.
a b ----- = ----- sin A sin B b(sin A) sin B = -------- a Plug in any known information.
25(sin 47[SUP]o[/SUP]) sin B = ----------- 15 Use a calculator to find the sine.
25(.7314) sin B = --------- 15 sin B = 1.219 Since an angle cannot have a sine greater than 1, there is no solution for this triangle.
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