To understand in which triangle the value of [tex]\( x \)[/tex] is equal to [tex]\( \tan^{-1} \left( \frac{3.1}{5.2} \right) \)[/tex], let's proceed with the following steps:
### Step 1: Understanding the Inverse Tangent Function
The inverse tangent function, denoted as [tex]\( \tan^{-1} \)[/tex] or [tex]\( \arctan \)[/tex], gives us an angle whose tangent is a given number. Specifically, if
[tex]\[ \tan(x) = \frac{3.1}{5.2}, \][/tex]
then
[tex]\[ x = \tan^{-1} \left( \frac{3.1}{5.2} \right). \][/tex]
### Step 2: Calculating the Value
Given that [tex]\(\frac{3.1}{5.2}\)[/tex] is a ratio of the opposite side to the adjacent side in a right triangle, we use [tex]\( \arctan \)[/tex]:
[tex]\[
x = \tan^{-1} \left( \frac{3.1}{5.2} \right).
\][/tex]
The result of this calculation is approximately:
[tex]\[ x \approx 0.5376 \text{ radians}. \][/tex]
### Step 3: Determining the Triangle Dimensions
To determine which triangle corresponds to this value, we need to ensure that the triangle has sides 3.1 and 5.2, with the angle [tex]\( x \)[/tex] such that:
[tex]\[
\tan(x) = \frac{3.1}{5.2}.
\][/tex]
This relationship tells us we should look for a right triangle where one angle (let’s call it [tex]\( \theta \)[/tex]) satisfies [tex]\( \theta = x \approx 0.5376 \text{ radians} \)[/tex].
### Step 4: Identifying the Triangle
Here's how you can identify the required triangle:
1. Right-Angle Triangle:
- One angle is [tex]\( x \approx 0.5376 \text{ radians} \)[/tex].
- The opposite side to this angle is 3.1.
- The adjacent side to this angle is 5.2.
So, the right triangle should have the following properties:
- Opposite side = 3.1 units.
- Adjacent side = 5.2 units.
- Hypotenuse can be calculated but is not directly needed for identifying the triangle contextually.
### Step 5: Conclusion
The required triangle is a right triangle where:
- One angle [tex]\( x \approx 0.5376 \text{ radians} \)[/tex] satisfies [tex]\( \tan(x) = \frac{3.1}{5.2} \)[/tex].
- The sides are consistent with this ratio [tex]\( \frac{\text{opposite}}{\text{adjacent}} = \frac{3.1}{5.2} \)[/tex].
This triangle's characteristics make it unique for the values given, particularly where:
[tex]\[ x = \tan^{-1} \left( \frac{3.1}{5.2} \right) \approx 0.5376 \text{ radians}. \][/tex]
Such a triangle is thus correctly identified by its side lengths and the angle [tex]\( x \)[/tex] derived from their ratio.
