The given image shows a triangle $\triangle ABC$ with the following known values:

• $AB = 12$ cm
• $BC = 32$ cm
• $\angle ABC = 72^\circ$

We can use the Cosine Rule to determine the unknown side $AC$:

$AC^2 = AB^2 + BC^2 - 2(AB)(BC) \cos(\angle ABC)$

Substituting the values:

$AC^2 = 12^2 + 32^2 - 2(12)(32) \cos(72^\circ)$

$AC^2 = 144 + 1024 - 2(12)(32) \cos(72^\circ)$

$AC^2 = 144 + 1024 - 768 \cos(72^\circ)$

Using $\cos 72^\circ \approx 0.309$,

$AC^2 = 144 + 1024 - 768(0.309)$

$AC^2 = 144 + 1024 - 237.312$

$AC^2 = 930.688$

$AC \approx \sqrt{930.688}$

$AC \approx 30.52 \text{ cm}$

Final Answer:

$AC \approx 30.52 \text{ cm}$

Would you like a detailed explanation of any step?

Related Questions:

1. How do you derive the Cosine Rule?
2. How can the Sine Rule be used to find another angle in this triangle?
3. How do you find the area of triangle $\triangle ABC$?
4. What are the possible errors when using trigonometric functions in calculations?
5. How can the Law of Sines help in solving this problem?

Tip:

Always check your calculator's angle mode (degrees/radians) when working with trigonometric functions!