Question

lucy started the proof of the law of cosines using triangle \\(wxy\\) as shown.

given: \\(\overline{xz} \perp \overline{wy}\\), \\(\delta wxz\\) and \\(\delta 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?

step 1 - the cosine ratio was applied incorrectly
step 2 - the sine ratio was applied incorrectly
step 3 - the incorrect value was subtracted
step 4 - the area formula should have been used instead.

Explanation:

Analyze Step 1

Using the Right Triangle Trigonometry knowledge point

$$
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Analyze Step 2

Using the Right Triangle Trigonometry knowledge point

$$
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The given step states: \(\sin(W) = \frac{y}{XZ}\) and \(XZ \sin(W) = y\), which is incorrect.

Analyze Step 3

Using the Law of Cosines knowledge point

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Analyze Step 4

Using the Law of Cosines knowledge point

$$
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Answer:

• Step 1 - the cosine ratio was applied incorrectly
• Step 2 - the sine ratio was applied incorrectly (Correct answer)
• Step 3 - the incorrect value was subtracted
• Step 4 - the area formula should have been used instead