Math Problem Statement

answer using sine rule

Determine the value of x using the Sine Rule based on the triangle provided.

Solution

To solve for xx in the triangle using the Sine Rule:

The Sine Rule states:

asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

From the diagram:

  1. The given sides and angles are:

    • Side opposite 38=13cm\text{Side opposite } 38^\circ = 13 \, \text{cm}
    • Side opposite 100=x\text{Side opposite } 100^\circ = x
  2. Let's apply the Sine Rule: xsin100=13sin38\frac{x}{\sin 100^\circ} = \frac{13}{\sin 38^\circ}

Rearrange to solve for xx: x=13sin100sin38x = \frac{13 \cdot \sin 100^\circ}{\sin 38^\circ}

Calculation:

  1. Compute sin100\sin 100^\circ and sin38\sin 38^\circ:

    • sin1000.9848\sin 100^\circ \approx 0.9848
    • sin380.6157\sin 38^\circ \approx 0.6157
  2. Substitute into the formula: x=130.98480.615712.80240.615720.8cmx = \frac{13 \cdot 0.9848}{0.6157} \approx \frac{12.8024}{0.6157} \approx 20.8 \, \text{cm}

Final Answer:

x20.8cmx \approx 20.8 \, \text{cm}

Let me know if you want a more detailed explanation or have further questions!

Related Questions:

  1. How does the Sine Rule compare to the Cosine Rule in solving triangles?
  2. What other scenarios can the Sine Rule be applied to?
  3. How do we calculate the third side of the triangle using the available information?
  4. How can you verify the answer using a different method?
  5. What are the implications of rounding sine values to fewer decimal places?

Tip:

Always double-check angle measurements in degrees versus radians when using trigonometric functions!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Rule

Formulas

Sine Rule: a/sin(A) = b/sin(B) = c/sin(C)

Theorems

Sine Rule

Suitable Grade Level

Grades 9-11