sympy import symbols, solve, Eq, sin, radians

Defining variables

WY = symbols('WY') # Length to solve for XW = 43 # Given angle_XWY = 72 # Angle at W angle_WXY = 69 # Angle at X

The sum of angles in a triangle is 180°. Solve for angle WYX

angle_WYX = 180 - angle_XWY - angle_WXY

Using the sine rule: XW / sin(WYX) = WY / sin(XWY)

Rearrange the formula to find WY

sin_rule = Eq(XW / sin(radians(angle_WYX)), WY / sin(radians(angle_XWY)))

Solving for WY

solution = solve(sin_rule, WY) solutionIt seems I encountered an issue with the library. Let me reattempt this with manual corrections for trigonometric angles.

Solution Explanation

We are tasked to find WY using the sine rule in triangle XWY. Here is the process step-by-step:

1. Identify Triangle Angles:

   • $\angle XWY = 72^\circ$ (given)
   • $\angle WXY = 69^\circ$ (given)
   • Using the sum of angles in a triangle:
     $\angle WYX = 180^\circ - 72^\circ - 69^\circ = 39^\circ$

2. Use the Sine Rule:
   The sine rule states: $\frac{\text{side opposite to } \angle WYX}{\sin(\angle WYX)} = \frac{\text{side opposite to } \angle XWY}{\sin(\angle XWY)}$ Here:

   • $XW = 43$ (opposite $\angle WYX$),
   • $WY$ is opposite $\angle XWY$.

   Substituting values: $\frac{43}{\sin(39^\circ)} = \frac{WY}{\sin(72^\circ)}$

3. Solve for WY:
   Rearranging: $WY = \frac{43 \cdot \sin(72^\circ)}{\sin(39^\circ)}$

Let me calculate the exact value now. math

Given values

XW = 43 angle_WYX = 39 # in degrees angle_XWY = 72 # in degrees

Applying the sine rule

WY = (XW * math.sin(math.radians(angle_XWY))) / math.sin(math.radians(angle_WYX)) WY### Final Answer: The length of $WY$ is approximately 65 units when rounded to the nearest integer.

Let me know if you would like more details or explanations!

Related Questions:

1. How can we derive the sine rule for triangles?
2. What is the importance of the sum of angles in a triangle?
3. How would the solution change if the angles or side lengths were different?
4. Can you solve for $WZ$ in the given triangle?
5. What would happen if the triangle was not a right triangle?

Tip: Always ensure angles are in degrees (or radians) when applying trigonometric functions.