11. In $$ riangle V W X, v=85 \mathrm{~cm}, x=58 \mathrm{~cm} $$ and $$ \angle X=107^{\circ} $$. Find all possible values of $$ \angle V $$, to the nearest degree.

Question

11. In $$ 	riangle V W X, v=85 \mathrm{~cm}, x=58 \mathrm{~cm} $$ and $$ \angle X=107^{\circ} $$. Find all possible values of $$ \angle V $$, to the nearest degree.

UpStudy AI · Accepted Answer

To find the possible values of $$ \angle V $$ in triangle $$ \triangle V W X $$ given the sides $$ v = 85 \, \text{cm} $$, $$ x = 58 \, \text{cm} $$, and the angle $$ \angle X = 107^\circ $$, we can use the Law of Sines.

The Law of Sines states that:

$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

In our case, we can denote:
- $$ a = v = 85 \, \text{cm} $$
- $$ b = x = 58 \, \text{cm} $$
- $$ C = \angle X = 107^\circ $$
- $$ A = \angle V $$
- $$ B = \angle W $$

Using the Law of Sines, we can write:

$$
\frac{v}{\sin V} = \frac{x}{\sin X}
$$

Substituting the known values:

$$
\frac{85}{\sin V} = \frac{58}{\sin 107^\circ}
$$

Now, we can solve for $$ \sin V $$:

$$
\sin V = \frac{85 \cdot \sin 107^\circ}{58}
$$

Next, we need to calculate $$ \sin 107^\circ $$ and then find $$ \sin V $$. Let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\sin\left(107\right)$$
Calculate or simplify the expression $$ (85 * 0.184782) / 58 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(85\times 0.184782\right)}{58}$$
- step1: Remove the parentheses:
  $$\frac{85\times 0.184782}{58}$$
- step2: Multiply the numbers:
  $$\frac{15.70647}{58}$$
- step3: Convert the expressions:
  $$\frac{\frac{1570647}{100000}}{58}$$
- step4: Multiply by the reciprocal:
  $$\frac{1570647}{100000}\times \frac{1}{58}$$
- step5: Multiply the fractions:
  $$\frac{1570647}{100000\times 58}$$
- step6: Multiply:
  $$\frac{1570647}{5800000}$$
We have calculated $$ \sin 107^\circ \approx 0.184782 $$. Now, substituting this value into our equation for $$ \sin V $$:

$$
\sin V = \frac{85 \cdot 0.184782}{58} \approx 0.270801
$$

Next, we need to find $$ \angle V $$ by taking the inverse sine:

$$
V = \sin^{-1}(0.270801)
$$

Let's calculate $$ V $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\arcsin\left(0.270801\right)$$
Calculate or simplify the expression $$ \arcsin(0.270801) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\arcsin\left(0.270801\right)$$
Calculate or simplify the expression $$ 0.274225 * (180 / \pi) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.274225\times \frac{180}{\pi }$$
- step1: Convert the expressions:
  $$\frac{10969}{40000}\times \frac{180}{\pi }$$
- step2: Multiply the fractions:
  $$\frac{10969\times 180}{40000\pi }$$
- step3: Multiply the terms:
  $$\frac{1974420}{40000\pi }$$
- step4: Reduce the fraction:
  $$\frac{98721}{2000\pi }$$
We found that:

$$
V \approx 15.71^\circ
$$

However, since the sine function is positive in both the first and second quadrants, there is a possibility for another angle $$ V' $$:

$$
V' = 180^\circ - V \approx 180^\circ - 15.71^\circ \approx 164.29^\circ
$$

Thus, the possible values for $$ \angle V $$ to the nearest degree are:

1. $$ \angle V \approx 16^\circ $$
2. $$ \angle V' \approx 164^\circ $$

In conclusion, the possible values of $$ \angle V $$ are $$ 16^\circ $$ and $$ 164^\circ $$.
The possible values of $$ \angle V $$ are $$ 16^\circ $$ and $$ 164^\circ $$.

11. In \( \triangle V W X, v=85 \mathrm{~cm}, x=58 \mathrm{~cm} \) and \( \angle X=107^{\circ} \). Find all possible values of \( \angle V \), to the nearest degree.

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