Question

Please help me I have to solve the triangle with the law of sines

Answer 1

Check the picture below.

and since a triangle has a sum of 180° for all interior angles.

A = 180 - 39.34 - 25

A = 115.66

$\bf \cfrac{sin(25^o)}{44}=\cfrac{sin(A)}{a}\implies \cfrac{sin(25^o)}{44}=\cfrac{sin(115.66^o)}{a} \\\\\\ a\cdot sin(25^o)=44\cdot sin(115.66^o)\implies a=\cfrac{44\cdot sin(115.66^o)}{sin(25^o)} \\\\\\ a\approx 93.8452\implies \stackrel{\textit{rounded up}}{a = 93.85}$

make sure your calculator is in Degree mode.

Answer 2

Answers

Angle B = 39.3 degrees OR Angle B = 140.7 degrees

Angle A = 115.7 degrees OR Angle A = 14.3 degrees

side a = 93.8 OR side a = 25.8

side note: there are two triangles possible which is why there are two answers per box

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Work Shown:

The lowercase letters are the sides, which are oppposite the uppercase letter angles. Eg: side b is opposite angle B

b = 66, c = 44, C = 25

Use the law of sines to determine angle B

sin(B)/b = sin(C)/c

sin(B)/66 = sin(25)/44

sin(B) = 66*sin(25)/44

sin(B) = 0.63392739261104

B = arcsin(0.63392739261104) or B = 180-arcsin(0.63392739261104)

B = 39.340476504461 or B = 180-39.340476504461

B = 39.340476504461 or B = 140.65952349554

B = 39.3 or B = 140.7

If angle B is 39.3, then angle A must be...

A+B+C = 180

A + 39.3 + 25 = 180

A + 64.3 = 180

A = 180 - 64.3

A = 115.7

Use this to find the length of side 'a'. Use the law of sines.

sin(A)/a = sin(C)/c

sin(115.7)/a = sin(25)/44

44*sin(115.7) = a*sin(25)

a*sin(25) = 44*sin(115.7)

a = 44*sin(115.7)/sin(25)

a = 93.8137144733656

a = 93.8

This triangle has the following sides and angles

sides: a = 93.8, b = 66, c = 44

angles: A = 115.7, B = 39.3, C = 25

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Repeat the same steps as above, but use B = 140.6595 instead

A+B+C = 180

A+140.6595+25 = 180

A+165.6595 = 180

A = 180-165.6595

A = 14.3405

A = 14.3

law of sines

sin(A)/a = sin(C)/c

sin(14.3405)/a = sin(25)/44

44*sin(14.3405) = a*sin(25)

a*sin(25) = 44*sin(14.3405)

a = 44*sin(14.3405)/sin(25)

a = 25.78708

a = 25.8

This triangle has the following sides and angles

sides: a = 25.8, b = 66, c = 44

angles: A = 14.3, B = 140.7, C = 25