ANSWERS
• m∠B = 59°
,
• m∠C = 81°
,
• c = 230.5 cm
EXPLANATION
Let's draw a diagram of this triangle first,
Given:
• a = 150 cm
,
• b = 200 cm
,
• m∠A = 40°
Find c, C and B.
First, with the given information, we can apply the law of sines to find B,
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
For this problem, we have to use the first two fractions,
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}[/tex]
Replace with the known values,
[tex]\frac{150\operatorname{cm}}{\sin40}=\frac{200\operatorname{cm}}{\sin B}[/tex]
To solve for B, first raise both sides of the equation to -1 - in other words, flip the fractions,
[tex]\frac{\sin40}{150\operatorname{cm}}=\frac{\sin B}{200\operatorname{cm}}[/tex]
Multiply both sides by 200cm,
[tex]\begin{gathered} \frac{\sin40}{150\operatorname{cm}}\cdot200cm=\frac{\sin B}{200\operatorname{cm}}\cdot200\operatorname{cm} \\ \frac{200\operatorname{cm}}{150\operatorname{cm}}\sin 40=\sin B \end{gathered}[/tex]
And use the inverse of the sine to find B,
[tex]B=\sin ^{-1}(\frac{200\operatorname{cm}}{150\operatorname{cm}}\sin 40)[/tex]
The three equations you can't see above are in the following picture,
Solve with a calculator: B = 59°
To find the third angle C, we know that the interior angles of any triangle add up to 180°. Thus, for this triangle,
• A + B + C = 180°
Solve for C,
• C = 180° - A - B
Replace with the values
• C = 180° - 40° - 59°
• C = 81 °
Then, using the law of sines again, we can find the length of side c.
[tex]\frac{a}{\sin A}=\frac{c}{\sin C}[/tex]
Solve for c and replace with the values,
[tex]c=a\cdot\frac{\sin C}{\sin A}=150\operatorname{cm}\cdot\frac{\sin81}{\sin40}=230.5\operatorname{cm}[/tex]
The two equations you can't see above are in the following picture,
The length of side c is 230.5 cm
