Question

given: $overline{xz}perpoverline{wy}$, $\triangle wxz$ and $\triangle yxz$ are right triangles
step 1: $cos(w)=\frac{wz}{y}$; $ycos(w)=wz$
step 2: $sin(w)=\frac{y}{xz}$; $xzsin(w)=y$
step 3: $zy = x - ycos(w)$
step 4: $w^{2}=(x - ycos(w))^{2}+xz^{2}$
which step contains the first error in calculation?
a. step 1 - the cosine ratio was applied incorrectly
b. step 2 - the sine ratio was applied incorrectly
c. step 3 - the incorrect value was subtracted
d. step 4 - the area formula should have been used instead.

Explanation:

Step1: Recall cosine ratio in right - triangle

In right - triangle $\triangle WXZ$, $\cos(W)=\frac{WZ}{y}$ (adjacent side over hypotenuse), so step 1 is correct.

Step2: Recall sine ratio in right - triangle

In right - triangle $\triangle WXZ$, the sine of an angle $W$ is $\sin(W)=\frac{XZ}{y}$ (opposite side over hypotenuse), not $\sin(W)=\frac{y}{XZ}$. So step 2 has an incorrect application of the sine ratio.

Step3: Analyze step 3

Since $WY = x$ and $WZ=y\cos(W)$, then $ZY=x - y\cos(W)$, step 3 is correct.

Step4: Analyze step 4

In right - triangle $\triangle XZY$, by the Pythagorean theorem $w^{2}=(x - y\cos(W))^{2}+XZ^{2}$, step 4 is correct.

Answer:

B. Step 2 - the sine ratio was applied incorrectly