Given triangle ABC where = 29cm, = 67o and = 45o. Find the length of the side , giving your answerto 3 significant figures

To find the length of side $AC$ in triangle $ABC$, we can use the Law of Sines. The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Given:

• $AB = 29$ cm
• $\angle B = 67^\circ$
• $\angle C = 45^\circ$

First, we need to find $\angle A$. Since the sum of the angles in a triangle is $180^\circ$:

$\angle A = 180^\circ - \angle B - \angle C$

$\angle A = 180^\circ - 67^\circ - 45^\circ$

$\angle A = 68^\circ$

Now, we can use the Law of Sines to find $AC$. Let $AC = b$, $AB = c = 29$ cm, and $\angle A = 68^\circ$, $\angle B = 67^\circ$.

$\frac{b}{\sin B} = \frac{c}{\sin C}$

$\frac{b}{\sin 67^\circ} = \frac{29}{\sin 45^\circ}$

$b = \frac{29 \cdot \sin 67^\circ}{\sin 45^\circ}$

Using the values of the sines:

$\sin 67^\circ \approx 0.9205$ $\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071$

$b = \frac{29 \cdot 0.9205}{0.7071}$

$b \approx \frac{26.6935}{0.7071}$

$b \approx 37.75$

Therefore, the length of side $AC$ is approximately $37.8$ cm to 3 significant figures.