Certainly! Let's solve the equation [tex]\(\cos^{-1}\left(\frac{3.4}{10}\right) = x\)[/tex] to determine the measure of angle BAC.
### Step-by-Step Solution
1. Fraction Simplification:
- First, simplify the fraction [tex]\(\frac{3.4}{10}\)[/tex]:
[tex]\[
\frac{3.4}{10} = 0.34
\][/tex]
- So, [tex]\(\cos^{-1}(0.34) = x\)[/tex].
2. Inverse Cosine Calculation:
- To find [tex]\(x\)[/tex], we need to calculate the inverse cosine (also known as arccosine) of 0.34. Using a calculator or a table of cosines:
[tex]\[
x \approx 1.2238794292677349 \text{ radians}
\][/tex]
- This means that [tex]\(x \approx 1.2238794292677349\)[/tex] radians when you take the arccosine of 0.34.
3. Conversion to Degrees:
- Often, angles are needed in degrees, so we convert the angle from radians to degrees.
- The conversion formula from radians to degrees is:
[tex]\[
\text{degrees} = x \times \left(\frac{180}{\pi}\right)
\][/tex]
- Substituting [tex]\(x \approx 1.2238794292677349\)[/tex] radians into the formula:
[tex]\[
x \approx 1.2238794292677349 \times \left(\frac{180}{\pi}\right) \approx 70.12312592992117 \text{ degrees}
\][/tex]
Hence, the measure of angle BAC is approximately [tex]\(70.12\)[/tex] degrees.
