A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

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Answer 1 · 2016-10-20T20:44:39

The largest area is $A \approx 0.68$ $A ~~ 0.68$

Explanation:

Let $∠ A = \frac{π}{6}$ $/_ A = pi/6$ and $∠ B = \frac{π}{4}$ $/_ B = pi/4$ , then $∠ C = π - \frac{π}{4} - \frac{π}{6} = \frac{7 π}{12}$ $/_C = pi - pi/4 - pi/6 = (7pi)/12$

Note: We assign the given side to be the side opposite the smallest angle, because that give the largest area.

Let a = the side opposite $∠ A$ $/_A$ = 1
Let b = the side opposite $∠ B$ $/_B$
Let c = the side opposite $∠ C$ $/_C$

We can use The Law of Sines to write the following equation:

$\frac{b}{sin (∠ B)} = \frac{a}{sin (∠ A)}$ $b/sin(/_B) = (a)/sin(/_A)$

Solve for b:

$b = (a) \frac{sin (∠ B)}{sin (∠ A)}$ $b = (a)sin(/_B)/sin(/_A)$

Choose side $a$ $a$ to be the base, then the height, h, of the triangle is:

$h = b sin (∠ C)$ $h = bsin(/_C)$

Substituting for b:

$h = ((a) \frac{sin (∠ B)}{sin (∠ A)}) sin (∠ C)$ $h = ((a)sin(/_B)/sin(/_A))sin(/_C)$

The area, A, with side a as the base is:

$A = \frac{1}{2} a h$ $A = 1/2ah$

Substituting for h:

$A = \frac{1}{2} a ((a) \frac{sin (∠ B)}{sin (∠ A)}) sin (∠ C)$ $A = 1/2a((a)sin(/_B)/sin(/_A))sin(/_C)$

$A = (a^{2}) \frac{sin (∠ B) sin (∠ C)}{2 sin (∠ A)}$ $A = (a^2)(sin(/_B)sin(/_C))/(2sin(/_A))$

$A = (1^{2}) \frac{sin (\frac{π}{4}) sin (\frac{7 π}{12})}{2 sin (\frac{π}{6})}$ $A = (1^2)(sin(pi/4)sin((7pi)/12))/(2sin(pi/6))$

$A \approx 0.68$ $A ~~ 0.68$

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{4}$ $pi / 4$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $1$ $1$ , what is the largest possible area of the triangle?