Evaluating Trigonometric Expression with Cosine Functions

The problem asks us to find the value of the expression: \[ \cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right) \] To solve this, we can use trigonometric identities to simplify the terms.

Applying Product-to-Sum Identity

Let's first simplify the last term, which is in the form of a product of two cosine functions: \(2 \cos A \cos B\). We use the product-to-sum identity: \[ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) \] Here, let \(A = \frac{11 \pi}{17}\) and \(B = \frac{\pi}{17}\).

Calculating the sum and difference of the angles:

• Sum of angles: \(A+B = \frac{11 \pi}{17} + \frac{\pi}{17} = \frac{11 \pi + \pi}{17} = \frac{12 \pi}{17}\)
• Difference of angles: \(A-B = \frac{1{1 \pi}}{17} - \frac{\pi}{17} = \frac{11 \pi - \pi}{17} = \frac{10 \pi}{17}\)

So, applying the identity:

Rewriting the Original Expression

Now, substitute this result back into the original expression:

Rearrange the terms for easier grouping:

Using Angle Relationships

Observe the angles in the expression. Notice that some angles are supplementary (sum up to \(\pi\)). We can use the property \(\cos(\pi - x) = -\cos x\).

• Consider the first pair: \(\cos \left(\frac{5 \pi}{17}\right) + \cos \left(\frac{12 \pi}{17}\right)\). The sum of the angles is \(\frac{5 \pi}{17} + \frac{12 \pi}{17} = \frac{17 \pi}{17} = \pi\). So, \(\frac{12 \pi}{17} = \pi - \frac{5 \pi}{17}\). Therefore, \(\cos \left(\frac{12 \pi}{17}\right) = \cos \left(\pi - \frac{5 \pi}{17}\right) = -\cos \left(\frac{5 \pi}{17}\right)\). The first pair becomes: \(\cos \left(\frac{5 \pi}{17}\right) + \left(-\cos \left(\frac{5 \pi}{17}\right)\right) = 0\).
• Consider the second pair: \(\cos \left(\frac{7 \pi}{17}\right) + \cos \left(\frac{10 \pi}{17}\right)\). The sum of the angles is \(\frac{7 \pi}{17} + \frac{10 \pi}{17} = \frac{17 \pi}{17} = \pi\). So, \(\frac{10 \pi}{17} = \pi - \frac{7 \pi}{17}\). Therefore, \(\cos \left(\frac{10 \pi}{17}\right) = \cos \left(\pi - \frac{7 \pi}{17}\right) = -\cos \left(\frac{7 \pi}{17}\right)\). The second pair becomes: \(\cos \left(\frac{7 \pi}{17}\right) + \left(-\cos \left(\frac{7 \pi}{17}\right)\right) = 0\).

Calculating the Final Value

The original expression simplifies to the sum of the values of these two pairs:

Thus, the value of the given trigonometric expression is 0.

Conclusion

By applying the product-to-sum identity and recognizing the relationship between angles summing to \(\pi\), we simplified the expression to find its value.

Revision Table: Key Trigonometric Identities

Additional Information: Understanding Angles in Radians

Angles in trigonometric functions are often expressed in radians. A full circle is \(2\pi\) radians, which is equivalent to 360 degrees. Half a circle is \(\pi\) radians, equivalent to 180 degrees. When working with angles like \(\frac{5\pi}{17}\), we are dealing with fractions of \(\pi\), representing a specific position on the unit circle. Understanding these relationships helps in simplifying trigonometric expressions and using identities like \(\cos(\pi - x) = -\cos x\).

The property \(\cos(\pi - x) = -\cos x\) arises because if an angle is \(x\), the angle \(\pi - x\) is its supplement. On the unit circle, these angles are symmetric with respect to the y-axis, resulting in cosine values that are equal in magnitude but opposite in sign.