Consider a right angled triangle where one of the non-hypotenuse sides has length 3 and the angle it makes with the hypotenuse is 65 degrees. What is the area of this triangle?

Steps to Solve

1. Identify the knowns and unknowns

We know one leg of the right triangle, let's call it $a = 5$. We also know the angle between this leg and the hypotenuse, $\theta = \frac{\pi}{12}$. We need to find the other leg, which we'll call $b$, to calculate the area.

2. Use trigonometry to find the other leg

Since we know the adjacent side ($a$) and the angle $\theta$, we can use the tangent function to find the opposite side ($b$):

$$ \tan(\theta) = \frac{b}{a} $$

Therefore:

$$ b = a \cdot \tan(\theta) $$

3. Substitute the given values

Substitute $a = 5$ and $\theta = \frac{\pi}{12}$ into the equation:

$$ b = 5 \cdot \tan\left(\frac{\pi}{12}\right) $$

4. Calculate $\tan(\frac{\pi}{12})$

We know that $\tan(\frac{\pi}{12}) = 2 - \sqrt{3}$. Therefore,

$$ b = 5 \cdot (2 - \sqrt{3}) $$

5. Calculate the area of the triangle

The area of a right triangle is given by:

$$ \text{Area} = \frac{1}{2} \cdot a \cdot b $$

Substitute $a = 5$ and $b = 5(2 - \sqrt{3})$:

$$ \text{Area} = \frac{1}{2} \cdot 5 \cdot 5(2 - \sqrt{3}) $$

$$ \text{Area} = \frac{25}{2} (2 - \sqrt{3}) $$