Question

Solve right triangle ABC for all missing parts. Express angles in decimal degrees.a =100.8 km, c=385.6 km We must find A, B, b

Answer 1

Given:

a = 100.8

c = 385.6

For angle "C"

It's a right angle triangle so

C = 90

For angle A is:

`\begin{gathered} b=\text{ base} \\ c=\text{ hypotenuse} \\ a=\text{ perpendicular} \end{gathered}`
`\begin{gathered} \sin A=\frac{\text{ perpendicular}}{\text{ hypotenuse}} \\ \sin A=(a)/(c) \\ \sin A=(100.8)/(385.6) \\ \sin A=0.2614 \\ A=\sin ^(-1)(0.2614) \\ A=15.1537 \end{gathered}`

For Angle B is:

`\begin{gathered} a=\text{ base} \\ c=\text{ hypotenuse} \\ b=\text{perpendicular} \end{gathered}`
`\begin{gathered} \cos B=\frac{\text{ base}}{\text{ hypotenuse}} \\ \cos B=(a)/(c) \\ \cos B=(100.8)/(385.6) \\ \cos B=0.2614 \\ B=\cos ^(-1)(0.2614) \\ B=74.8462 \end{gathered}`

For "b" use pythagoras theorem.

`\begin{gathered} (\text{hypotenuse)}^2=(\text{base)}^2+(\text{perpendicular)}^2 \\ c^2=b^2+a^2 \\ (385.6)^2=b^2+(100.8)^2 \\ 148687.36=b^2+10160.64 \\ b^2=148687.36-10160.64 \\ b^2=138526.72 \\ b=\sqrt[]{138526.72} \\ b=372.1917 \end{gathered}`