Question

Calculate the values of a, A and C in triangle ABC given that b = 17.23cm , c= 10.86cm and B = 101°15'

Answer 1

Given Side Lengths:

a = ?

b = 17.23

c = 10.86

Given Angles (in degrees):

A = ?

B = 101

C = ?

Note: We have highlighted the angle(s) and side that we need to find. Let's draw a rough sketch of the triangle:

The sin rule is:

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

From the information given, we can write a ratio using the sine rule:

`(\sin101)/(17.23)=(\sin C)/(10.86)`

We can cross multiply and solve for the value of angle C:

`\begin{gathered} (\sin101)/(17.23)=(\sin C)/(10.86) \\ 10.86\sin 101=17.23\sin C \\ \sin C=(10.86\sin101)/(17.23) \\ \sin C=0.6187 \\ C=\sin ^(-1)(0.6187) \\ C=38.22\degree \end{gathered}`

Now, we know 3 angles in a triangle add up to 180 degrees. Thus, we can solve for the Angle A:

`\begin{gathered} A+B+C=180 \\ A+101+38.22=180 \\ A+139.22=180 \\ A=180-139.22 \\ A=40.78\degree \end{gathered}`

We can write another ratio using the sin rule and find the side length, a. Shown below:

`(\sin101)/(17.23)=(\sin40.78)/(a)`

Let's do a little algebra and solve for the side length, a:

`\begin{gathered} (\sin101)/(17.23)=(\sin40.78)/(a) \\ a\sin 101=17.23\sin 40.78 \\ a=(17.23\sin 40.78)/(\sin 101) \\ a=11.46 \end{gathered}`Answer
`\begin{gathered} a=11.46\text{ cm} \\ A=40.78\degree \\ C=38.22\degree \end{gathered}`