A triangle has two corners with angles of # pi / 12 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #3 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  pi  /  12  # and # (5 pi  )/  8   #.  If one side of the triangle has a length of #3  #, what is the largest possible area of the triangle?

Evelyn Arboleda · Accepted Answer

#color(maroon)("Largest possible Area of triangle " A_t = 12.5 " sq units"#

#hat A = pi/12, hat B = (5pi) / 8, hat C = pi - pi/12 - (5pi) / 8 = (7pi)/24#

To get the largest area, side 3 should correspond to least angle #hatA#

Applying the Law of Sines,

#a / sin A = b / sin B = c / sin C#

#3 / sin (pi/12) = b / sin ((5pi)/8)#

#b = (3 * sin ((5pi)/8)) / sin (pi/12) = 10.71#

#"Area of Triangle " A_t = (1/2) a b sin C#

#A_t = (1/2) * 3 * 10.71 * sin ((7pi)/24) = 12.5 " sq units"#

Noah Atkinson · Answer

undefined To find the largest possible area of the triangle given two angles $ \frac{\pi}{12} $ and $ \frac{5\pi}{8} $ and one side length of 3, we can use the formula for the area of a triangle:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Let's denote the two given angles as $ \alpha = \frac{\pi}{12} $ and $ \beta = \frac{5\pi}{8} $. We'll consider $ \alpha $ as the angle between the given side of length 3 and the base of the triangle.

Using trigonometry, we can find the height of the triangle in terms of $ \alpha $ and the given side length:

$$ \text{Height} = 3 \times \tan(\alpha) $$

Now, we can express the area of the triangle in terms of $ \alpha $:

$$ \text{Area}(\alpha) = \frac{1}{2} \times 3 \times 3 \times \tan(\alpha) $$

To find the maximum area, we need to maximize $ \text{Area}(\alpha) $ with respect to $ \alpha $. We can do this by taking the derivative of $ \text{Area}(\alpha) $ with respect to $ \alpha $, setting it equal to zero, and solving for $ \alpha $.

$$ \frac{d}{d\alpha} \left( \frac{9}{2} \tan(\alpha) \right) = 0 $$

Solving this equation will give us the value of $ \alpha $ that maximizes the area of the triangle. Once we have $ \alpha $, we can find the corresponding value of $ \beta $, and then calculate the area of the triangle using the given side length and the derived angles.