Calculating the Cosine Sum: cos 20° + cos 100° + cos 140°

This solution explains how to find the value of the trigonometric expression cos 20° + cos 100° + cos 140°.

Understanding the Expression

We need to calculate the sum of the cosines of three different angles: 20 degrees, 100 degrees, and 140 degrees. This problem can be solved using trigonometric identities, specifically the sum-to-product formulas.

Applying the Sum-to-Product Formula

The sum-to-product formula for cosine is:

$$ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) $$

Let's apply this formula to the terms cos 100° and cos 140° in the expression.

Let $ A = 140^\circ $ and $ B = 100^\circ $. Then:

• $ \frac{A+B}{2} = \frac{140^\circ + 100^\circ}{2} = \frac{240^\circ}{2} = 120^\circ $
• $ \frac{A-B}{2} = \frac{140^\circ - 100^\circ}{2} = \frac{40^\circ}{2} = 20^\circ $

Substituting these into the formula:

$$ \cos 100^\circ + \cos 140^\circ = 2 \cos(120^\circ) \cos(20^\circ) $$

Evaluating Intermediate Terms

We know the value of $ \cos(120^\circ) $. The angle 120° lies in the second quadrant, where cosine is negative.

$$ \cos(120^\circ) = \cos(180^\circ - 60^\circ) = -\cos(60^\circ) = -\frac{1}{2} $$

Now substitute this value back into the sum-to-product result:

$$ \cos 100^\circ + \cos 140^\circ = 2 \left(-\frac{1}{2}\right) \cos(20^\circ) $$

$$ \cos 100^\circ + \cos 140^\circ = -1 \cdot \cos(20^\circ) = -\cos(20^\circ) $$

Final Calculation

Now, let's substitute this result back into the original expression:

Original Expression = $ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ $

Substitute the simplified part:

Expression = $ \cos 20^\circ + (-\cos 20^\circ) $

Expression = $ \cos 20^\circ - \cos 20^\circ $

Expression = $ 0 $

Conclusion

The value of the expression cos 20° + cos 100° + cos 140° is 0.