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Side A has a length of 12 m, and angle Y is 12°. What is the length of side B (in m)?

Question

Side A has a length of 12 m, and angle Y is 12°. What is the length of side B (in m)?

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Solution

1. Break Down the Problem

To find the length of side B in a triangle where we know the length of side A and angle Y, we can use the sine function. Given:

  • Length of side A = 12 m
  • Angle Y = 12°

Side B can be calculated using the sine law, which relates the lengths of sides of a triangle to the sines of the angles opposite those sides.

2. Relevant Concepts

Using the sine rule, we have: Asin(Y)=Bsin(X) \frac{A}{\sin(Y)} = \frac{B}{\sin(X)} where:

  • A A is the length of side A (12 m)
  • B B is the length of side B we want to find
  • Y Y is angle Y (12°)
  • X X is the angle opposite side B, which we need additional information to determine.

If we assume a right triangle for simplicity and knowing one angle, we can directly use calculations without needing X X .

3. Analysis and Detail

In a right triangle where Y Y is small, we would typically consider A A opposite to angle Y Y :

Using the sine of angle Y Y : sin(12)0.2079 \sin(12^\circ) \approx 0.2079 From the sine definition: A=Bsin(Y)    B=Asin(Y)=12sin(12) A = B \cdot \sin(Y) \implies B = \frac{A}{\sin(Y)} = \frac{12}{\sin(12^\circ)}

Calculating B B : B=120.207957.742 B = \frac{12}{0.2079} \approx 57.742

4. Verify and Summarize

Calculation of B B : B57.742 m B \approx 57.742 \text{ m} This result was obtained through the sine rule, confirming a relationship in the triangle.

Final Answer

The length of side B is approximately 57.74 m.

This problem has been solved

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