To solve for the value of [tex]\( x \)[/tex] where [tex]\( x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex], we need to understand the process involved in finding the angle using the inverse cosine function. Let's go through it step-by-step:
1. Identify the Given Values:
- We are given the ratio [tex]\(\frac{4.3}{6.7}\)[/tex].
2. Understanding [tex]\(\cos^{-1}\)[/tex] Function:
- The inverse cosine function, [tex]\(\cos^{-1}(y)\)[/tex], gives us the angle whose cosine is [tex]\( y \)[/tex].
- In this case, [tex]\(\cos x = \frac{4.3}{6.7}\)[/tex].
3. Calculation of [tex]\(\frac{4.3}{6.7}\)[/tex]:
- We simplify the fraction first.
- [tex]\(\frac{4.3}{6.7} \approx 0.641791\)[/tex].
4. Apply the Inverse Cosine Function:
- Find [tex]\( x \)[/tex] such that [tex]\( x = \cos^{-1}(0.641791) \)[/tex].
5. Calculate the Angle:
- Using a calculator, we find [tex]\( x \approx 0.8739648401891128 \)[/tex] radians.
6. Convert Radians to Degrees (if needed):
- To convert from radians to degrees, we use the formula: [tex]\(\text{degrees} = \text{radians} \times \frac{180}{\pi}\)[/tex].
- Hence, [tex]\( x_{\text{degrees}} = 0.8739648401891128 \times \frac{180}{\pi} \approx 50.07449678566164 \)[/tex] degrees.
Thus, the angle [tex]\( x \)[/tex] in radians is [tex]\( 0.8739648401891128 \)[/tex] and in degrees is [tex]\( 50.07449678566164 \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] in the triangle, where [tex]\( \cos x = \frac{4.3}{6.7} \)[/tex], is [tex]\( 0.8739648401891128 \)[/tex] radians or [tex]\( 50.07449678566164 \)[/tex] degrees.
