To determine which triangle corresponds to the given value of [tex]\( x \)[/tex], we need to follow these steps:
1. Understand the Function: The expression [tex]\(\cos^{-1}\left(\frac{4.3}{6.7}\right)\)[/tex] is the inverse cosine function. This function will give us the angle [tex]\( x \)[/tex] whose cosine is [tex]\(\frac{4.3}{6.7}\)[/tex].
2. Calculate the Cosine Ratio: Before we determine the angle [tex]\( x \)[/tex], let's recognize that [tex]\(\frac{4.3}{6.7}\)[/tex] is the ratio of the lengths of the adjacent side to the hypotenuse in a right triangle.
3. Find the Angle [tex]\( x \)[/tex]: By evaluating [tex]\(\cos^{-1}\left(\frac{4.3}{6.7}\right)\)[/tex], we find [tex]\( x \)[/tex]. The result from calculations gives us:
[tex]\[
x \approx 0.873965 \text{ radians}
\][/tex]
This can also be converted to degrees if necessary:
[tex]\[
x \approx 50.07^\circ
\][/tex]
4. Identify the Triangle: Given this angle [tex]\( x \)[/tex], we are looking for a triangle where one of the angles is approximately [tex]\(50.07^\circ\)[/tex]. Additionally, the cosine of this angle should match the given ratio, [tex]\(\frac{4.3}{6.7}\)[/tex].
Let's check the properties of different triangles:
- Triangle with [tex]\( x \approx 50.07^\circ\)[/tex]: This triangle has:
- Adjacent side = 4.3 units
- Hypotenuse = 6.7 units
In this triangle, the cosine of the angle [tex]\( x \)[/tex] would be [tex]\(\frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4.3}{6.7}\)[/tex].
5. Verify the Triangle:
- The given cosine value matches the ratio [tex]\(\frac{4.3}{6.7}\)[/tex].
- The given angle [tex]\( x \)[/tex] is approximately [tex]\( 50.07^\circ \)[/tex].
Therefore, the correct triangle corresponding to the given value of [tex]\( x \)[/tex] is the one where:
- The triangle’s angle [tex]\( x \)[/tex] is approximately [tex]\(50.07^\circ\)[/tex].
- The lengths of the sides match the ratio of 4.3 (adjacent) to 6.7 (hypotenuse).
So, the triangle with one angle of approximately [tex]\( 50.07^\circ \)[/tex] and sides in the ratio [tex]\( \frac{4.3}{6.7} \)[/tex] is the triangle that corresponds to [tex]\( x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex].
