Question

assume α is opposite side a, β is opposite side b, and γ is opposite side c. solve the triangle. round all answers to 1 decimal place, where necessary.
α = 78°, γ = 21°, a = 9
β =
b =
c =

Explanation:

Step1: Find angle $\beta$

The sum of angles in a triangle is $180^{\circ}$. So, $\beta=180^{\circ}-\alpha - \gamma$.
Substituting $\alpha = 78^{\circ}$ and $\gamma=21^{\circ}$, we get $\beta=180^{\circ}-78^{\circ}-21^{\circ}=81^{\circ}$.

Step2: Use the Law of Sines to find side $b$

The Law of Sines is $\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}$.
We know $a = 9$, $\alpha=78^{\circ}$, $\beta = 81^{\circ}$. Then $b=\frac{a\sin\beta}{\sin\alpha}$.
$\sin78^{\circ}\approx0.9781$, $\sin81^{\circ}\approx0.9877$.
$b=\frac{9\times0.9877}{0.9781}=\frac{8.8893}{0.9781}\approx9.1$.

Step3: Use the Law of Sines to find side $c$

By the Law of Sines $\frac{a}{\sin\alpha}=\frac{c}{\sin\gamma}$.
$\sin21^{\circ}\approx0.3584$.
$c=\frac{a\sin\gamma}{\sin\alpha}=\frac{9\times0.3584}{0.9781}=\frac{3.2256}{0.9781}\approx3.3$.

Answer:

$\beta = 81^{\circ}$, $b\approx9.1$, $c\approx3.3$