Verified Answer :
- Recognize that the equation \(\sin(x) = \frac{5}{8.3}\) defines angle \(x\) in a right-angled triangle.
- Identify that 5 is the opposite side and 8.3 is the hypotenuse of this triangle.
- Use the Pythagorean theorem to find the third side length \(\sqrt{8.3^2 - 5^2} = \sqrt{43.89} \approx 6.625\).
- Conclude that the condition is met by any right-angled triangle with side lengths in ratio \(5 : \sqrt{43.89} : 8.3\).
- Final answer: \(\boxed{\text{Any right-angled triangle with side ratio } 5 : \sqrt{43.89} : 8.3 \text{ where } 5 \text{ is opposite to } x}\)
Explanation
- Analyze the equation defining angle \(x\).
The problem asks us to define the condition under which an angle \(x\) in a triangle satisfies the equation \(x = \sin^{-1}\left(\frac{5}{8.3}\right)\). This equation is equivalent to saying that the sine of the angle \(x\) is \(\sin(x) = \frac{5}{8.3}\). In any right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Thus, if we consider a right-angled triangle with an angle \(x\), the side opposite to angle \(x\) must have a length of 5 and the hypotenuse must have a length of 8.3.
- Calculate the third side using the Pythagorean theorem.
To determine the properties of such a right-angled triangle, we use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. We know that the side opposite to \(x\) is \(a = 5\) and the hypotenuse is \(c = 8.3\). We can solve for the adjacent side \(b\):
\[5^2 + b^2 = 8.3^2\]
\[25 + b^2 = 68.89\]
\[b^2 = 68.89 - 25\]
\[b^2 = 43.89\]
\[b = \sqrt{43.89} \approx 6.625\]
Therefore, in a right-angled triangle where the leg opposite to angle \(x\) is 5 and the hypotenuse is 8.3, the adjacent leg must be \(\sqrt{43.89} \approx 6.625\).
- Summarize the triangle properties.
The condition \(x = \sin^{-1}\left(\frac{5}{8.3}\right)\) is satisfied by any right-angled triangle whose sides are in the ratio \(5 : \sqrt{43.89} : 8.3\), where the side of length 5 is opposite to the angle \(x\) and the side of length 8.3 is the hypotenuse. This defines the family of triangles that have the specified angle measure. The value of the angle itself is approximately \(\arcsin(0.6024) \approx 37.04^\circ\). The final answer is that such an angle \(x\) occurs in any right-angled triangle where the ratio of the opposite side to the hypotenuse is \(\frac{5}{8.3}\).
Examples
Imagine you are a carpenter designing a support beam that must slope at a specific angle to fit into a wall corner. If you need the slope angle to be \(x = \sin^{-1}\left(\frac{5}{8.3}\right)\), you can measure 5 units vertically (the rise) and 8.3 units along the diagonal (the hypotenuse) of your beam. By ensuring these proportions in your right-angled design, you guarantee the support beam meets the wall at exactly the required angle, ensuring structural stability and a perfect fit.