A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/4# and the angle between sides B and C is #pi/6#. If side B has a length of 2, what is the area of the triangle?

Using the triangle and the sine rule

The triangle's area is

To find the area of the triangle with sides (A), (B), and (C) given the lengths of sides (B) and the angles between them, we can use the law of sines to find the lengths of the other sides and then apply the formula for the area of a triangle.

Given:

• (B = 2)
• Angle between sides (A) and (B) is (\frac{\pi}{4}) (45 degrees)
• Angle between sides (B) and (C) is (\frac{\pi}{6}) (30 degrees)

Using the law of sines: [ \frac{A}{\sin(\frac{\pi}{6})} = \frac{2}{\sin(\frac{\pi}{4})} ] Solving for (A): [ A = 2 \cdot \frac{\sin(\frac{\pi}{6})}{\sin(\frac{\pi}{4})} ]

Similarly: [ \frac{C}{\sin(\frac{\pi}{6})} = \frac{2}{\sin(\frac{\pi}{6})} ] Solving for (C): [ C = 2 \cdot \frac{\sin(\frac{\pi}{6})}{\sin(\frac{\pi}{6})} ]

Now, we can use Heron's formula to find the area of the triangle: [ s = \frac{A + B + C}{2} ] [ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} ]

Substituting the values of (A), (B), and (C), we get: [ s = \frac{A + 2 + C}{2} ] [ \text{Area} = \sqrt{s(s - A)(s - 2)(s - C)} ]

Calculate (s) and substitute the values into the formula to find the area of the triangle.