In triangle △ABC, ∠BAC=75∘, ∠ACB=60∘, and AB=82. Find BC.
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Step-by-Step Solution
Step 1: Find the third angle of the triangle
The sum of the angles in any triangle is always 180∘. We are given two angles, ∠BAC=75∘ and ∠ACB=60∘. We can find the third angle, ∠ABC, by subtracting the sum of the given angles from 180∘.
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, ∠BAC. We also know the length of side AB and the angle opposite to it, ∠ACB.
Step 3: Substitute known values into the Sine Rule
Now we substitute the given values into the Sine Rule equation. We have ∠BAC=75∘, AB=82, and ∠ACB=60∘.
Step 4: Solve for BC
To find BC, we isolate it by multiplying both sides of the equation by sin(75∘). We use the known values for sin(75∘)=46+2 and sin(60∘)=23 and simplify the expression.