https://file.aiquickdraw.com/mathbot/file/17276038632964mn5q5mq.jpg

The image shows a triangle with two sides labeled \(6.7 \, \text{cm}\) and \(5.4 \, \text{cm}\), along with an angle of \(95^\circ\) between them. We are tasked with finding the angle \(x^\circ\) opposite to the side of length \(5.4 \, \text{cm}\), and the answer should be given to 3 significant figures.

This can be solved using the **sine rule**. The sine rule is written as:

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

Where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite those sides.

### Step-by-step solution:
1. Label the triangle: 
   - \(a = 6.7 \, \text{cm}\), opposite the unknown angle \(x^\circ\).
   - \(b = 5.4 \, \text{cm}\), opposite the angle \(95^\circ\).
   
2. Apply the sine rule to find \(x^\circ\):

\[
\frac{6.7}{\sin x} = \frac{5.4}{\sin 95^\circ}
\]

We can now solve this equation for \(x\).

Let me compute this.It seems there is an issue with the computation, likely due to the calculated value of \(\sin(x)\) exceeding the valid range (which should be between \(-1\) and \(1\)).

This suggests that I may have made a mistake in assuming the triangle is possible with these dimensions or setup. Let me verify the steps and recompute to ensure the values stay within valid limits.

The calculated value of \(\sin(x)\) exceeds 1, which is not possible, as the sine of any angle must lie between \(-1\) and \(1\). This suggests the given triangle dimensions are inconsistent, meaning such a triangle cannot exist with the provided lengths and angle.

Would you like to clarify the problem, or should I help explain further why this situation arises? 

Let me know if you'd like more details. Here are five related questions that could help expand on this problem:

1. How can we determine when a triangle's dimensions are impossible?
2. What is the sine rule, and how does it apply to solving triangles?
3. What steps are involved in validating triangle configurations?
4. How do rounding or measurement errors affect real-world problems like this one?
5. What are some common mistakes to watch out for when solving triangles with the sine rule?

### Tip:
Always ensure that the sum of two sides of a triangle is greater than the third side to validate its existence before applying any formulas.

Work out the value of x in a triangle with sides 6.7 cm and 5.4 cm, and an angle of 95 degrees between them. Give your answer to 3 significant figures.

Find out how to solve for an unknown angle in a triangle using the sine rule. This problem involves a triangle with sides 6.7 cm and 5.4 cm, and an angle of 95 degrees between them. Learn why certain triangle configurations may be impossible.

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