Question

Consider the triangle shown in the diagram below.Suppose that m∠A=80∘, m∠C=33∘, and c=45.1. What is the value of aa?a=Now, suppose that m∠B=48∘, m∠C=26∘, and a=19.9. What is the value of cc?c=

Answer 1

The Law of Sines

It's an equation that relates the lengths of the sides of a triangle with the sines of its angles.

For the given triangle, the equation is:

`(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)`

a.

We are given: m/_A = 80°, m/_C = 33°, and c=45.1.

We use the first and the last part of the equation above:

`\begin{gathered} (a)/(\sin A)=(c)/(\sin C) \\ (a)/(\sin80^o)=(45.1)/(\sin 33^o) \end{gathered}`

Solving for a:

`a=\sin 80^o\cdot(45.1)/(\sin33^o)`

Calculating:

`a=0.9848\cdot(45.1)/(0.5446)=81.6`

a = 81.6

b.

Now we are given m/_B=48°, m/_C=26°, a=19.9

Since we are required to calculate the value of c and we are not given the value of the angle A, we first determine it recalling the sum of angles of a triangle is 180°, thus:

m/_A= 180° - 48° - 26° = 106°

Now we apply the equation:

`\begin{gathered} (a)/(\sin A)=(c)/(\sin C) \\ (19.9)/(\sin 106^o)=(c)/(\sin 26^o) \end{gathered}`

Solving for c:

`\begin{gathered} c=\sin 26^o\cdot(19.9)/(\sin106^o) \\ c=0.4384\cdot(19.9)/(0.9613)=9.08 \end{gathered}`

c = 9.08