To solve the expression [tex]\(\cos 40^\circ \cdot \cos 100^\circ \cdot \cos 160^\circ\)[/tex], we can employ trigonometric identities and the properties of cosine functions for specific angles. Let's break it down step-by-step:
### Step 1: Identify the angles
The angles in the given expression are [tex]\(40^\circ\)[/tex], [tex]\(100^\circ\)[/tex], and [tex]\(160^\circ\)[/tex]. Notice that:
- [tex]\(40^\circ + 100^\circ + 160^\circ = 300^\circ \neq 180^\circ\)[/tex], so they do not form a complete set of supplementary angles. However, we know some trigonometric properties about the involved angles.
### Step 2: Utilize trigonometric identities
We can use the known identity for cosine for angles that sum up to multiples of [tex]\(180^\circ\)[/tex]. We have the following identities:
- [tex]\(\cos(180^\circ - \theta) = -\cos(\theta)\)[/tex]
Using the above identity, re-write some of the angles:
- [tex]\(100^\circ = 180^\circ - 80^\circ \implies \cos 100^\circ = -\cos 80^\circ\)[/tex]
- [tex]\(160^\circ = 180^\circ - 20^\circ \implies \cos 160^\circ = -\cos 20^\circ\)[/tex]
### Step 3: Rewrite the expression with the known identities
Using the rewritten cosine functions, we have:
[tex]\[
\cos 40^\circ \cdot \cos 100^\circ \cdot \cos 160^\circ = \cos 40^\circ \cdot (-\cos 80^\circ) \cdot (-\cos 20^\circ)
\][/tex]
Since the product of two negative cosines will be positive, we get:
[tex]\[
\cos 40^\circ \cdot \cos 100^\circ \cdot \cos 160^\circ = \cos 40^\circ \cdot \cos 80^\circ \cdot \cos 20^\circ
\][/tex]
### Step 4: Use product-to-sum identities if needed
There are advanced trigonometric identities that help evaluate the product of cosines directly. In this case, we recognize that the expression [tex]\(\cos 40^\circ \cdot \cos 80^\circ \cdot \cos 20^\circ\)[/tex] can be directly related to a known trigonometric product identity for specific angles:
[tex]\[
\cos 40^\circ \cdot \cos 80^\circ \cdot \cos 20^\circ = \frac{1}{8}
\][/tex]
### Conclusion
Therefore, the value of [tex]\(\cos 40^\circ \cdot \cos 100^\circ \cdot \cos 160^\circ\)[/tex] is indeed:
[tex]\[
\boxed{\frac{1}{8}}
\][/tex]
When you further verify this numerically, it is confirmed that:
[tex]\[
\cos 40^\circ \cdot \cos 100^\circ \cdot \cos 160^\circ = 0.125
\][/tex]
Thus, the original trigonometric product simplifies to [tex]\(\frac{1}{8}\)[/tex], confirming the detailed analysis and identities used.
