Question

law of sines
two right triangles, △abd and △acd, were created from △abc by constructing (overline{ad} perp overline{cb}).
follow these steps to prove that (\frac{sin(b)}{b} = \frac{sin(c)}{c}).

1. for △acd, (sin(c) = ) dropdown
2. multiply both sides of the equation by (b) and simplify. the equation (sin(c) = \frac{h}{b}) becomes dropdown
3. by the dropdown property, we can set the expressions equal: something = (b sin(c)) (options: multiplication, substitution)

check.
diagram: triangle abc with (ad perp cb), labels a, b, c, d, (h), (b), (c)

Explanation:

Question 3:

Step1: Recall sine definition in right triangle

In right triangle \( \triangle ACD \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle C \), opposite side is \( h \), hypotenuse is \( b \).
So \( \sin(C) = \frac{h}{b} \).

Step1: Multiply both sides by \( b \)

Given \( \sin(C) = \frac{h}{b} \), multiply both sides by \( b \): \( b \times \sin(C) = b \times \frac{h}{b} \).

Step2: Simplify the right side

\( b \times \frac{h}{b} = h \), so the equation becomes \( b\sin(C) = h \).

We have two expressions for \( h \): from \( \triangle ABD \), we would have \( h = c\sin(B) \) (similar to how we got \( h = b\sin(C) \) from \( \triangle ACD \)). To set \( c\sin(B) = b\sin(C) \), we use the substitution property (since both equal \( h \), we substitute one for the other). The multiplication property doesn't apply here; substitution is used to replace \( h \) with its equivalent expression.

Answer:

\( \frac{h}{b} \)

Question 4: