To determine which triangle has the value of [tex]\( x \)[/tex] such that [tex]\( x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex], let's follow a detailed, step-by-step approach:
1. Understand the relationship: The expression [tex]\( \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex] refers to the angle whose cosine value is [tex]\( \frac{4.3}{6.7} \)[/tex].
2. Calculate the ratio:
[tex]\[
\frac{4.3}{6.7} \approx 0.6418
\][/tex]
3. Determine angle [tex]\( x \)[/tex]: The inverse cosine function, [tex]\( \cos^{-1} \)[/tex], gives us an angle whose cosine is the given ratio. Let's denote this angle by [tex]\( x \)[/tex].
4. Find the measure of angle [tex]\( x \)[/tex]:
[tex]\[
x \approx \cos^{-1}(0.6418) \approx 0.874 \text{ radians}
\][/tex]
5. Convert radians to degrees: To better interpret this angle in a triangle, we convert it from radians to degrees,
[tex]\[
x \approx 0.874 \text{ radians} \times \left(\frac{180}{\pi}\right) \approx 50.074 \text{ degrees}
\][/tex]
6. Interpretation: Now, we know that the angle [tex]\( x \)[/tex] in the triangle is approximately [tex]\( 50.074 \)[/tex] degrees.
In summary:
- The ratio of the sides is [tex]\( \frac{4.3}{6.7} \approx 0.6418 \)[/tex].
- The angle [tex]\( x \)[/tex] corresponding to [tex]\( \cos^{-1}(0.6418) \)[/tex] is approximately [tex]\( 0.874 \)[/tex] radians.
- Converting [tex]\( 0.874 \)[/tex] radians to degrees, [tex]\( x \approx 50.074 \)[/tex] degrees.
Thus, in the triangle, the value of [tex]\( x \)[/tex] is approximately [tex]\( 50.074 \)[/tex] degrees. The triangle where one of the angles is [tex]\( 50.074 \)[/tex] degrees is the one that we're looking for.
