Question

I need help to match these can someone help me.

Answer 1

Answer: See attached image below

Explanation:

The law of sines is

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

This is the same as writing

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Because we apply the reciprocal to each side to keep things balanced

Note how the 'a's stick together on one side, so do the 'b's and 'c's.

For something like $\frac{a}{b}\sin B$ we can see the $\frac{\sin B}{b}$ portion, but now there's an 'a' in there. This is from multiplying both sides of the equation below by 'a'

$\frac{\sin A}{a} = \frac{\sin B}{b}\\\\\sin A= a\frac{\sin B}{b}\\\\\sin A= \frac{a}{b}\sin B$

The summary is that the 'a' pairs with sin(A) and that's what matches with expression number 1.

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For expression number 2, we'll have it pair with sin(B). This is because the 'c's already are taken care of (they are paired up) and there's an extra b all by itself. The working/steps will basically look like what is shown in the section above, though you'll use the equation $\frac{\sin B}{b} = \frac{\sin C}{c}$

Expression 3 will match with sin(C) for similar reasons as expression 2. The 'c' is the only letter that isn't paired up.

The other expressions will match in similar fashion. The filled out answers are shown in the attached image below.