Question

the law of cosines is used to find the measure of ∠y.
18² = 16² + 19² - 2(16)(19)cos(y)
324 = 256 + 361 - (608)cos(y)
324 = 617 - (608)cos(y)
-293 = -608cos(y)
to the nearest whole degree, what is the measure of ∠y?
57°
41°
47°
51°

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines formula for an angle \(Y\) in a triangle with sides \(a\), \(b\), \(c\) opposite to angles \(A\), \(B\), \(C\) respectively is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In our case, let's assume the sides: from the triangle, let \(XY = 16\), \(Y\) to the other vertex (let's say \(Z\)) is \(19\), and \(XZ\) is \(18\) (from the formula \(18^{2}=16^{2}+19^{2}-2(16)(19)\cos(Y)\)). So we have the equation \(18^{2}=16^{2}+19^{2}-2\times16\times19\times\cos(Y)\).

Step2: Calculate the squares

Calculate \(18^{2}=324\), \(16^{2} = 256\), \(19^{2}=361\). Substitute into the equation: \(324=256 + 361-608\cos(Y)\).

Step3: Simplify the right - hand side

First, add \(256\) and \(361\): \(256 + 361=617\). So the equation becomes \(324=617-608\cos(Y)\).

Step4: Rearrange the equation to solve for \(\cos(Y)\)

Subtract \(617\) from both sides: \(324 - 617=-608\cos(Y)\). Calculate \(324 - 617=-293\). So we have \(-293=-608\cos(Y)\).

Step5: Solve for \(\cos(Y)\)

Divide both sides by \(- 608\): \(\cos(Y)=\frac{-293}{-608}=\frac{293}{608}\approx0.4819\).

Step6: Find the angle \(Y\)

Take the inverse cosine of \(0.4819\): \(Y=\cos^{-1}(0.4819)\approx47^{\circ}\) (rounded to the nearest whole degree).

Answer:

\(47^{\circ}\)