A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{5 π}{12}$ $( 5 pi)/12$ , what are the lengths of sides A and B?

Question

A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{5 π}{12}$ $( 5 pi)/12$ , what are the lengths of sides A and B?

1 Answer

Impact of this question

1547 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-22T15:16:14

side $a = \sqrt{6} + \sqrt{2} = 3.863703305$ $a=sqrt(6)+sqrt(2)=3.863703305$
side $b = \sqrt{6} + \sqrt{2} = 3.863703305$ $b=sqrt(6)+sqrt(2)=3.863703305$

Explanation:

The given parts of the triangle are
side $c = 2$ $c=2$
Angle $A = \frac{5 π}{12} = 75^{\circ}$ $A=(5pi)/12=75^@$
Angle $C = \frac{π}{6} = 30^{\circ}$ $C=pi/6=30^@$

The third angle $B$ $B$ can be readily solved

$A + B + C = 180^{\circ}$ $A+B+C=180^@$

$75^{\circ} + B + 30^{\circ} = 180^{\circ}$ $75^@+B+30^@=180^@$

$B = 180 - 105^{\circ} = 75^{\circ}$ $B=180-105^@=75^@$

Therefore we have an isosceles triangle with angles A and B equal.

solve side $a$ $a$ using Sine Law

$\frac{a}{sin A} = \frac{c}{sin C}$ $a/sin A=c/sin C$

$a = \frac{c \cdot sin A}{sin C} = \frac{2 \cdot {sin 75}^{\circ}}{{sin 30}^{\circ}} = \sqrt{6} + \sqrt{2}$ $a=(c*sin A)/sin C=(2*sin 75^@)/(sin 30^@)=sqrt(6)+sqrt(2)$
by the definition of Isosceles triangle

$b = a = \sqrt{6} + \sqrt{2}$ $b=a=sqrt(6)+sqrt(2)$

We can check triangle using the Mollweide's Equation which involves all the 6 parts of the triangle

$\frac{a - c}{b} = \frac{sin (\frac{1}{2} (A - C))}{cos (\frac{1}{2} (B))}$ $(a-c)/b=sin (1/2(A-C))/(cos (1/2(B)))$

$\frac{\sqrt{6} + \sqrt{2} - 2}{\sqrt{6} + \sqrt{2}} = \frac{sin (\frac{1}{2} (75 - 30))}{cos (\frac{1}{2} (75))}$ $(sqrt(6)+sqrt(2)-2)/(sqrt(6)+sqrt(2))=sin (1/2(75-30))/(cos (1/2(75)))$

$0.4823619098 = 0.4823619098$ $0.4823619098=0.4823619098$

This means all the 6 parts are correct.

God bless....I hope the explanation is useful.

Science

Math

Humanities

... and beyond

A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{5 π}{12}$ $( 5 pi)/12$ , what are the lengths of sides A and B?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A triangle has sides A, B, and C. The angle between sides A and B is π6(pi)/6. If side C has a length of 22 and the angle between sides B and C is 5π12( 5 pi)/12, what are the lengths of sides A and B?

1 Answer

Explanation:

Related questions

Impact of this question

A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $2$ $2$ and the angle between sides B and C is $\frac{5 π}{12}$ $( 5 pi)/12$ , what are the lengths of sides A and B?