General category: Mathematics.
Sub-category: Triangles.
Topic: Law of Sines
Introduction:
The Law of Sines is the relationship between the sides and angles of oblique triangles. This law states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.
Explanation:
In a triangle ABC, the Law of Sines tell us the following expression:
[tex]\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}[/tex]
Consider the following oblique triangle:
Here:
CB= a
CA= b
AB= c
∠B= 180 - (32 +30)= 180 - 62 = 118
Now, applying the law of sines, we get the following expression:
[tex]\frac{CB}{\sin(30\degree)}=\frac{30}{\sin(118\degree)}=\frac{AB}{\sin(32\degree)}[/tex]
Then, we get the following equations:
Equation 1:
[tex]\frac{CB}{\sin(30\degree)}=\frac{30}{\sin(118\degree)}[/tex]
Equation 2:
[tex]\frac{30}{\sin(118\degree)}=\frac{AB}{\sin(32\degree)}[/tex]
From equation 1, we obtain:
[tex]a=CB=\frac{30\cdot\text{ }\sin(30\degree)}{\sin(118\degree)}=\text{ 16.98855}\approx17[/tex]
From equation 2, we get:
[tex]c=AB=\frac{30\cdot\text{ }\sin(32\degree)}{\sin(118\degree)}=18.00512\approx18[/tex]
We can conclude that the correct answer is:
Answer:
m∠B=118
a= 17
c= 18
