Question

find the length of side c. c a 120° b 3 c a 15° b = 45° a = 1.1 c = ?

Explanation:

Step1: Identify the Law of Sines

In a triangle, the Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Here, we know angle $A = 15^\circ$, angle $B = 45^\circ$, side $a = 1.1$, and we need to find side $c$. First, we can find angle $C$ using the fact that the sum of angles in a triangle is $180^\circ$. So, $C = 180^\circ - 15^\circ - 45^\circ = 120^\circ$, which matches the given angle at $C$.

Step2: Apply the Law of Sines for side \( c \)

Using the Law of Sines, $\frac{c}{\sin C} = \frac{a}{\sin A}$. We know $a = 1.1$, $\sin A = \sin 15^\circ$, $\sin C = \sin 120^\circ$. So we can solve for $c$: $c = \frac{a \cdot \sin C}{\sin A}$.

Step3: Calculate the sines

We know that $\sin 15^\circ \approx 0.2588$ and $\sin 120^\circ = \sin (60^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660$.

Step4: Substitute the values

Substitute $a = 1.1$, $\sin C \approx 0.8660$, and $\sin A \approx 0.2588$ into the formula for $c$: $c = \frac{1.1 \times 0.8660}{0.2588}$.

Step5: Perform the calculation

First, calculate the numerator: $1.1 \times 0.8660 = 0.9526$. Then divide by the denominator: $c = \frac{0.9526}{0.2588} \approx 3.7$.

Answer:

\( 3.7 \)