Question

select the correct answer.
lucy started the proof of the law of cosines using triangle wxy as shown.
given: \\(\overline{xz} \perp \overline{wy}\\), \\(\triangle wxz\\) and \\(\triangle yxz\\) are right triangles
step 1: \\(\cos(w) = \frac{wz}{y}\\); \\(y\cos(w) = wz\\)
step 2: \\(\sin(w) = \frac{y}{xz}\\); \\(xz\sin(w) = y\\)
step 3: \\(zy = x - y\cos(w)\\)
step 4: \\(w^2 = (x - y\cos(w))^2 + xz^2\\)
which step contains the first error in calculation?
\\(\bigcirc\\) a. step 1 – the cosine ratio was applied incorrectly
\\(\bigcirc\\) b. step 2 – the sine ratio was applied incorrectly
\\(\bigcirc\\) c. step 3 – the incorrect value was subtracted
\\(\bigcirc\\) d. step 4 – the area formula should have been used instead.

Explanation:

Step1: Check Step 1

In right triangle \(WXZ\), \(\cos(W) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{WZ}{y}\), so \(y\cos(W) = WZ\). Correct.

Step2: Check Step 2

In right triangle \(WXZ\), \(\sin(W) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{XZ}{y}\), so \(XZ = y\sin(W)\). Step 2 incorrectly wrote \(\sin(W) = \frac{y}{XZ}\) and \(XZ\sin(W) = y\). Error here.

Step3: Confirm first error

Step 2 is the first step with an error.

Answer:

B. Step 2 – the sine ratio was applied incorrectly