Sure! Let's work through this problem step-by-step together.
The problem involves finding the value of [tex]\( x \)[/tex] in a triangle such that:
[tex]\[ x = \tan^{-1}\left(\frac{3.1}{5.2}\right) \][/tex]
The [tex]\(\tan^{-1}\)[/tex] function, also known as the arctangent function, gives the angle whose tangent is the given ratio. In this case, the ratio is [tex]\(\frac{3.1}{5.2}\)[/tex].
1. Understand the given ratio:
- The tangent of an angle [tex]\( \theta \)[/tex] in a right triangle is defined as the ratio of the opposite side to the adjacent side.
- Here, the ratio is [tex]\(\frac{3.1}{5.2}\)[/tex], which means if you have a right triangle, the length of the side opposite angle [tex]\( x \)[/tex] is 3.1 units, and the length of the side adjacent to angle [tex]\( x \)[/tex] is 5.2 units.
2. Right Triangle Configuration:
- Consider a right triangle where one of the acute angles is [tex]\( x \)[/tex].
- For [tex]\( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex], we have:
- Opposite side = 3.1
- Adjacent side = 5.2
3. Finding the angle [tex]\( x \)[/tex]:
- From the given ratio, [tex]\( x \)[/tex] is the angle whose tangent is [tex]\(\frac{3.1}{5.2}\)[/tex].
- This angle can be evaluated using trigonometric tables or a calculator.
Based on the numerically evaluated answer, we have:
[tex]\[ x \approx 0.5376 \text{ radians} \][/tex]
Converting to degrees (if needed), since [tex]\( 1 \text{ radian} \approx 57.2958 \text{ degrees} \)[/tex]:
[tex]\[ x \approx 0.5376 \times 57.2958 \approx 30.8 \text{ degrees} \][/tex]
So, the triangle we are considering has the following configuration for angle [tex]\( x \)[/tex]:
- A right triangle.
- An angle [tex]\( x \approx 0.5376 \text{ radians} \)[/tex] or [tex]\( \approx 30.8 \text{ degrees} \)[/tex].
- The side opposite to [tex]\( x \)[/tex] is 3.1 units.
- The side adjacent to [tex]\( x \)[/tex] is 5.2 units.
We have identified the triangle based on the given ratio:
- A right triangle with sides 3.1 and 5.2 corresponding to the opposite and adjacent sides of angle [tex]\( x \)[/tex], respectively.
This detailed approach lets us identify the right triangle that fits the conditions provided in the problem statement.
