Question

let abc be a triangle such that angle a is opposite side a, angle b is opposite side b, and angle c is opposite side c. if ( b = 112.26^{circ}), ( a = 13), and ( b = 15), solve for angle a. round your answer to the nearest hundredth of a degree.

( mangle a=) type your answer... ( ^{circ})

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}$.
Substituting the given values: $\frac{13}{\sin A}=\frac{15}{\sin(112.26^{\circ})}$.

Step2: Solve for $\sin A$

Cross - multiply to get $15\sin A = 13\sin(112.26^{\circ})$.
First, calculate $\sin(112.26^{\circ})\approx0.925$.
Then $15\sin A=13\times0.925 = 12.025$.
So, $\sin A=\frac{12.025}{15}\approx0.8017$.

Step3: Find angle $A$

Since $\sin A\approx0.8017$, then $A=\sin^{- 1}(0.8017)\approx53.21^{\circ}$.

Answer:

$53.21$