Sure, let's find the measure of angle BAC step-by-step using the given equation:
The equation provided is:
[tex]\[
\cos^{-1}\left(\frac{3.4}{10}\right) = x
\][/tex]
Our goal is to determine the value of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] represents the angle measure in degrees.
1. Understand what the equation means: The expression [tex]\(\cos^{-1}(y)\)[/tex] (also written as [tex]\(\arccos(y)\)[/tex]) gives us the angle whose cosine is [tex]\( y \)[/tex]. In this case, [tex]\( y = \frac{3.4}{10} \)[/tex].
2. Calculate the value inside the inverse cosine function:
[tex]\[
\frac{3.4}{10} = 0.34
\][/tex]
3. Interpret the result of [tex]\(\cos^{-1}(0.34)\)[/tex]: Using the inverse cosine function, we find [tex]\( x \)[/tex] which is the angle whose cosine is 0.34. Using a calculator or inverse cosine tables, we find:
[tex]\[
x \approx 70.12312592992117 \text{ degrees}
\][/tex]
4. Round the angle to the nearest whole degree: Looking at the decimal part, we see it is closer to 70 than to 71. Therefore, rounding 70.12312592992117 to the nearest whole number gives us:
[tex]\[
x \approx 70 \text{ degrees}
\][/tex]
The degree measure of angle BAC, when rounded to the nearest whole degree, is [tex]\(\boxed{70^{\circ}}\)[/tex].
