Sure, let's solve the given expression step by step:
Given the expression:
[tex]\[ 8 \cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ} = ? \][/tex]
1. Identify the values of the cosine functions:
We are given [tex]\(\cos 10^{\circ}\)[/tex], [tex]\(\cos 50^{\circ}\)[/tex], and [tex]\(\cos 70^{\circ}\)[/tex].
2. Multiply them together:
Multiply the cosine values:
[tex]\[ \cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ} \][/tex]
3. Multiply by 8:
After we have the result from step 2, we then multiply this product by 8:
[tex]\[ 8 \cdot (\cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ}) \][/tex]
Upon performing these calculations accurately, you obtain a numerical result. For this specific expression, the numerical value is approximately [tex]\(1.7320508075688779\)[/tex].
To verify our result, let's compare it with the square root of 3:
4. Evaluate [tex]\(\sqrt{3}\)[/tex]:
The square root of 3 is approximately:
[tex]\[ \sqrt{3} \approx 1.7320508075688772 \][/tex]
Finally, when comparing the obtained product, [tex]\(1.7320508075688779\)[/tex], with [tex]\(\sqrt{3} \approx 1.7320508075688772\)[/tex], we see that they are essentially equal within typical computational uncertainty.
Thus, we can conclude that:
[tex]\[ 8 \cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ} = \sqrt{3} \][/tex]
