Question

In AVWX, v = 180 inches, w = 170 inches and 2X-86°. Find 2V, to the nearest degree.

Answer Attempt 2 out of 2
O
Submit Answer

Answer 1

Use the law of cosines to find VW, then use the law of cosines again to find the angle WVX. It just takes a lot of computation.

Answer 2

∠V = 49°

Explanation:

In triangle VWX:

• v = 180 inches
• w = 170 inches
• ∠X = 86°

As we have the measure of two sides (v and w) and the included angle (X), we can use the cosine rule to find the length of side x, then use the sine rule to find ∠V.

`\boxed{\begin{array}{l}\underline{\textsf{Cosine Rule}}\\\\c^2=a^2+b^2-2ab \cos C\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides.}\\\phantom{ww}\bullet\;\textsf{$C$ is the angle opposite side $c$.}\end{array}}`

So, in this case:

`x^2=v^2+w^2-2vw \cos X`

Substitute the values of v, w and X into the formula, and solve for the exact value of x:

`x^2=180^2+170^2-2(180)(170) \cos 86^(\circ)`

`x^2=61300-61200 \cos 86^(\circ)`

`x=\sqrt{61300-61200 \cos 86^(\circ)}`

Now, we can use the sine rule to find the measure of angle V.

`\boxed{\begin{array}{l}\underline{\textsf{Sine Rule}} \\\\(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}`

In this case:

`(\sin V)/(v)=(\sin W)/(w)=(\sin X)/(x)`

Substitute the values of v, x and X into the formula:

`(\sin V)/(180)=\frac{\sin 86^(\circ)}{\sqrt{61300-61200 \cos 86^(\circ)}}`

Solve for V:

`\sin V=\frac{180\sin 86^(\circ)}{\sqrt{61300-61200 \cos 86^(\circ)}}`

`V=\sin ^(-1)\left(\frac{180\sin 86^(\circ)}{\sqrt{61300-61200 \cos 86^(\circ)}}\right)`

`V=48.7549424...^(\circ)`

`V=49^(\circ)\sf \; (nearest\;degree)`

Therefore, the measure of ∠V is 49° (rounded to the nearest degree).