Using the law of sines and the law of cosines, the solutions for the triangle are given as follows:
What is the law of cosines?
The law of cosines states that we can find the side c of a triangle as follows:
c² = a² + b² - 2abcos(C)
In which:
- C is the angle opposite to side c.
- a and b are the lengths of the other sides.
For this problem, side c is found using the law of cosines, as follows:
c² = 67² + 62² - 2(67)(62)cos(114º)
c² = 11712.17
c = sqrt(11712.17)
c = 108.
What is the law of sines?
Suppose we have a triangle in which:
- The length of the side opposite to angle A is a.
- The length of the side opposite to angle B is b.
- The length of the side opposite to angle C is c.
The lengths and the sine of the angles are related as follows:
[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]
Angle A can be found as follows:
[tex]\frac{\sin{A}}{67} = \frac{\sin{114^\circ}}{108}[/tex]
[tex]\sin{A} = \frac{67\sin{114^\circ}}{108}[/tex]
sin(A) = 0.5667365339
A = arcsin(0.5667365339)
A = 34.52º.
The sum of the internal angles of a triangle is of 180º, hence we use it to find angle B as follows:
34.52 + B + 114 = 180
B = 180 - (34.52 + 114)
B = 31.48º.
More can be learned about the law of sines at https://brainly.com/question/25535771
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