Answer:
Step-by-step explanation:To solve this problem, we can use the Law of Sines, which relates the sides of a triangle to the sines of their opposite angles. The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given:
- \(AC = 14 \text{ cm}\)
- \(BC = 10 \text{ cm}\)
- \(\angle ABC = 63^\circ\)
We need to find:
1. \(\angle CAB\) (denoted as \(\angle A\))
2. The length of side \(AB\) (denoted as \(c\))
### Step 1: Find \(\angle A\) (using Law of Sines)
We know:
- \(b = AC = 14 \text{ cm}\)
- \(a = BC = 10 \text{ cm}\)
- \(\angle B = 63^\circ\)
Using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Substitute the known values:
\[
\frac{10}{\sin A} = \frac{14}{\sin 63^\circ}
\]
Now, solve for \(\sin A\):
\[
\sin A = \frac{10 \times \sin 63^\circ}{14}
\]
Calculate the value:
\[
\sin A = \frac{10 \times 0.8910}{14} \approx \frac{8.910}{14} \approx 0.636
\]
Now find \(\angle A\) (use the inverse sine function):
\[
\angle A \approx \sin^{-1}(0.636) \approx 39.5^\circ
\]
### Step 2: Find \(\angle C\)
We can find \(\angle C\) using the fact that the sum of the angles in a triangle is \(180^\circ\):
\[
\angle C = 180^\circ - \angle A - \angle B
\]
\[
\angle C = 180^\circ - 39.5^\circ - 63^\circ \approx 77.5^\circ
\]
### Step 3: Find the length of \(AB\) (using Law of Sines)
Using the Law of Sines again:
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
Substitute the known values:
\[
\frac{c}{\sin 77.5^\circ} = \frac{10}{\sin 39.5^\circ}
\]
Solve for \(c\):
\[
c = \frac{10 \times \sin 77.5^\circ}{\sin 39.5^\circ}
\]
Calculate the value:
\[
c = \frac{10 \times 0.9775}{0.6360} \approx \frac{9.775}{0.6360} \approx 15.37 \text{ cm}
\]
### Final Answers:
1. \(\angle CAB \approx 39.5^\circ\)
2. The length of \(AB\) is approximately \(15.37\) cm.
