Question

Complete the proof of the Law of Sines/Cosines.

Answer 1

Answer:

(A)

1. Definition of altitude
2. c⋅sinA
3. $c\cdot sinA=a\cdot sinC$.

Step-by-step explanation:

Law of Sines/Cosines.

• Given triangle ABC with altitude segment BD labeled x. Angles ADB and CDB are right angles by the (1) definition of altitude making triangle ABD and triangle BCD right triangles.

• Using the trigonometric ratios $sin A=\dfrac{x}{c}$ and $sin C=\dfrac{x}{a}$.

• Multiplying to isolate x in both equations gives (2)$x=c\cdot sinA$ and $x=a\cdot sinC.$

• We also know that x = x by the reflexive property.

• By the substitution property,(3)$c\cdot sinA=a\cdot sinC$.

• Dividing each side of the equation by ac gives:

$\frac{c\cdot sinA}{ac} =\frac{a\cdot sinC}{ac} \\\frac{ sinA}{a} =\frac{sinC}{c}$