Proving the Rational Value of 1 cos 40° cos 80° cos 160°

Understanding trigonometric identities and mathematical proofs can be both challenging and rewarding. One such interesting problem is to prove whether the sum 1 cos 40° cos 80° cos 160° is a rational value. In this article, we will go through the detailed steps to solve this problem using proven trigonometric identities and logical reasoning.

Step 1: Initial Expression

We start with the initial expression:

1 cos 40^circ cos 80^circ cos 160^circ

Step 2: Simplifying Using Trigonometric Identities

We use the double-angle identity for cosine, which states that:

1 cos 2theta 2cos^2 theta

Applying this identity, we can write:

1 cos 40^circ 2cos^2 20^circ 1 cos 80^circ 2cos^2 40^circ 1 cos 160^circ 2cos^2 80^circ

Let S denote the sum:

S 1 cos 40^circ cos 80^circ cos 160^circ

Substituting the values, we get:

S 2cos^2 20^circ 2cos^2 40^circ 2cos^2 80^circ

We factor out the common coefficient:

S 2 (cos^2 20^circ cos^2 40^circ cos^2 80^circ)

Step 3: Further Simplification

We use the identity:

cos 2theta 1 - 2sin^2 theta

to transform the terms:

Note that:

cos 40^circ 1 - 2sin^2 20^circ cos 80^circ 1 - 2sin^2 40^circ cos 160^circ -cos 20^circ - (1 - 2sin^2 10^circ)

Therefore, the expression for S becomes:

S 2 left(cos^2 20^circ (1 - 2sin^2 20^circ) (1 - 2sin^2 40^circ) (1 - 2sin^2 10^circ)right)

Combining like terms, we have:

S 2 left(4 - 2sin^2 20^circ - 2sin^2 40^circ - 2sin^2 10^circright)

Step 4: Simplifying Further

We simplify the expression by combining constants and terms involving sines:

S 8 - 4left(sin^2 20^circ sin^2 40^circ sin^2 10^circright)

Using trigonometric identities, we know that for any angle theta:

sin^2 theta frac{1 - cos 2theta}{2}

Applying this identity, we get:

sin^2 20^circ frac{1 - cos 40^circ}{2} sin^2 40^circ frac{1 - cos 80^circ}{2} sin^2 10^circ frac{1 - cos 20^circ}{2}

Substituting these into the expression for S:

S 8 - 4left(frac{1 - cos 40^circ}{2} frac{1 - cos 80^circ}{2} frac{1 - cos 20^circ}{2}right)

Simplifying, we get:

S 8 - 4left(frac{3 - (cos 40^circ cos 80^circ cos 20^circ)}{2}right)

Further simplifying:

S 8 - 2 (3 - (1 cos 40^circ cos 80^circ cos 160^circ))

Step 5: Final Simplification and Proof

Recall that the initial expression for S is:

S 1 cos 40^circ cos 80^circ cos 160^circ

Therefore, substituting the simplified form:

8 - 2 (3 - S) S

Solving for S:

8 - 6 2S S

Combining like terms:

2S S - 2 S frac{2}{8} frac{1}{4}

The result is:

S frac{1}{8}

Therefore, the sum 1 cos 40° cos 80° cos 160° is a rational value, and it is equal to 1/8.

Conclusion

In conclusion, by using trigonometric identities and logical reasoning, we have successfully proven that 1 cos 40° cos 80° cos 160° 1/8. This problem showcases the power of trigonometric identities and their applications in proving mathematical results.