In triangle $\triangle ABC$, $\angle BAC=75^\circ$, $\angle ACB=60^\circ$, and $AB=8\sqrt{2}$. Find $BC$.

Step 1: Find the third angle of the triangle

The sum of the angles in any triangle is always $180^\circ$. We are given two angles, $\angle BAC = 75^\circ$ and $\angle ACB = 60^\circ$. We can find the third angle, $\angle ABC$, by subtracting the sum of the given angles from $180^\circ$.

Step 2: Apply the Sine Rule

The Sine Rule states that in any triangle, the ratio of the length of a side to the sine of the angle opposite that side is constant. We want to find the length of side $BC$, and we know the angle opposite to it, $\angle BAC$. We also know the length of side $AB$ and the angle opposite to it, $\angle ACB$.

Step 3: Substitute known values into the Sine Rule

Now we substitute the given values into the Sine Rule equation. We have $\angle BAC = 75^\circ$, $AB = 8\sqrt{2}$, and $\angle ACB = 60^\circ$.

Step 4: Solve for BC

To find $BC$, we isolate it by multiplying both sides of the equation by $\sin(75^\circ)$. We use the known values for $\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$ and $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ and simplify the expression.

More from this chapter

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