A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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Answer 1 · 2018-03-18T00:59:10

$A = 2.93$ $A = 2.93$
$B = 5.66$ $B = 5.66$

Explanation:

Given two angles, the third one in a triangle is fixed. In this case it is $2 \frac{π}{12}$ $2pi/12$ . The shortest side length will be opposite the smallest angle, which is $\frac{π}{12}$ $pi/12$ in this case. We know that the side of length 8 is opposite the $9 \frac{π}{12}$ $9pi/12$ corner.

We now have three angles and a side, and can calculate the other sides using the Law of Sines, and then calculate the height for the area.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-sines

https://www.mathsisfun.com/algebra/trig-solving-asa-triangles.html

$\frac{a}{sin (\frac{π}{12})} = \frac{c}{sin C} = \frac{8}{sin (9 \frac{π}{12})}$ $a/(sin(pi/12)) = c/sin C = 8/(sin(9pi/12))$
$\frac{b}{sin (2 \frac{π}{12})} = \frac{c}{sin C} = \frac{8}{sin (9 \frac{π}{12})}$ $b/(sin(2pi/12)) = c/sin C = 8/(sin(9pi/12))$

$a \times (sin (9 \frac{π}{12})) = 8 \times (sin (\frac{π}{12}))$ $a xx (sin(9pi/12)) = 8 xx (sin(pi/12))$

$b \times (sin (9 \frac{π}{12})) = 8 \times (sin (2 \frac{π}{12}))$ $b xx (sin(9pi/12)) = 8 xx (sin(2pi/12))$

$a \times 0.707 = 8 \times 0.259$ $a xx 0.707 = 8 xx 0.259$ ; $a = 2.93$ $a = 2.93$
$b \times 0.707 = 8 \times 0.50$ $b xx 0.707 = 8 xx 0.50$ ; $b = 5.66$ $b = 5.66$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is 3π4(3pi)/4. If side C has a length of 88 and the angle between sides B and C is π12pi/12, what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?