Question

Solve for the remaining angles and side of the triangle described below. Round to the nearest thousandth:B = 100°, c = 5, a = 2

Answer 1

Given:

B = 100 degrees

c = 5

a = 2

We are interested in getting b, C and A

In any Given triangle ABC with sides a,b,c facing angles A,B,C respectively,

`\begin{gathered} b^2=a^2+c^2-2acCosB----------(co\sin e\text{ rule)} \\ \end{gathered}`

Thus,

`\begin{gathered} b^2=2^2+5^2-2(2)(5)Cos100 \\ b^2=4+25-20(-0.1736) \\ b^2=29+3.47 \\ b^2=32.47 \\ b=\sqrt[]{32.47} \\ b=5.6982 \\ b=5.698\text{ (nearest thousandth)} \end{gathered}`

To calculate angle A:

`\begin{gathered} \text{ Using sine rule, } \\ \frac{a}{\sin\text{ A}}=(b)/(\sin B) \\ \frac{2}{\sin\text{ A}}=(5.6982)/(\sin100) \\ \\ 5.6982\text{ x sinA=2 x sin 100} \\ \sin A=\frac{2\text{ x sin 100}}{5.6982} \\ \sin \text{ A=0.34566} \\ A=\sin ^(-1)0,34566 \\ A=20.2221 \\ A=20.222^0\text{ (nearest thousandth)} \end{gathered}`

To calculate angle C

`\begin{gathered} \text{Angle A + Angle B + Angle C = 180 degre}es\text{ ( Sum of angles in a triangle)} \\ 20.222+100_{}+C=180 \\ 120.222+C=180 \\ C=180-120.222 \\ C=59.778^0 \end{gathered}`