A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{8}$ $(5pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 3, what is the area of the triangle?

Question

A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{8}$ $(5pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 3, what is the area of the triangle?

1 Answer

Impact of this question

1746 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-06T16:01:19

Area $= 1.356 {units}^{2}$ $= 1.356" units" ^2$ to 3 decimal places

Explanation:

Tony B

$Method$ $color(blue)("Method")$

Find h using the sin rule. Then use h to determine the area.

$Solution$ $color(blue)("Solution")$

Target is to be able to apply $\frac{C}{sin (c)} = \frac{B}{sin (b)}$ $C/(sin(c))=B/(sin(b))$

To do this we need to find $∠ c b a$ $/_cba$

The sum of the internal angles in a triangle is $180^{o}$ $180^o$

$\Rightarrow ∠ c b a = π - \frac{5 π}{8} - \frac{π}{12} = \frac{7 π}{24} ({52.5}^{o})$ $=> /_cba=pi-(5pi)/8-(pi)/12=(7pi)/24" " (52.5 ^o)$

Thus we have

$\frac{C}{sin (\frac{5 π}{8})} = \frac{3}{sin (\frac{7 π}{24})}$ $C/(sin((5pi)/8)) = 3/(sin((7pi)/24))$

$C = 3 \times \frac{sin (\frac{5 π}{8})}{sin (\frac{7 π}{24})}$ $C= 3 xxsin((5pi)/8)/(sin((7pi)/24))$

$h = C sin (\frac{π}{12})$ $h= C sin(pi/12)$

$h = 3 \times \frac{sin (\frac{5 π}{8})}{sin (\frac{7 π}{24})} \times sin (\frac{π}{12})$ $h= 3 xxsin((5pi)/8)/(sin((7pi)/24))xxsin(pi/12)$

$A r e a = \frac{B}{2} \times h = \frac{9}{2} \times \frac{sin (\frac{5 π}{8})}{sin (\frac{7 π}{24})} \times sin (\frac{π}{12})$ $Area = B/2xxh" "=" "9/2 xxsin((5pi)/8)/(sin((7pi)/24))xxsin(pi/12)$

$\frac{9}{2} \times \frac{sin ({112.5}^{o})}{sin ({52.5}^{o})} \times sin (15^{o})$ $9/2xx(sin(112.5^o))/(sin(52.5^o))xxsin(15^o)$

$= 1.356 {units}^{2}$ $=1.356 " units"^2$

Science

Math

Humanities

... and beyond

A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{8}$ $(5pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 3, what is the area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A triangle has sides A,B, and C. If the angle between sides A and B is 5π8(5pi)/8, the angle between sides B and C is π12pi/12, and the length of B is 3, what is the area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{8}$ $(5pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 3, what is the area of the triangle?