Evaluating Trigonometric Products and Their Sum

The problem asks us to find the value of \(p+q\), where \(p\) and \(q\) are defined as products of cosine terms with specific angles.

The given expressions are:

• \( \rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \)
• \( \rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right) \)

Evaluating the Value of p

Let's evaluate the expression for \(p\):

\( \rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \)

We can use the trigonometric identity \(2 \sin A \cos A = \sin 2A\). To apply this, we can multiply and divide the expression by \(2 \sin \left(\frac{\pi}{5}\right)\):

\( \rm p = \frac{1}{2 \sin \left(\frac{\pi}{5}\right)} \left(2 \sin \left(\frac{\pi}{5}\right) \cos \left(\frac{\pi}{5}\right)\right) \cos \left(\frac{2 \pi}{5}\right) \)

Applying the identity to the term in the parenthesis:

\( \rm p = \frac{1}{2 \sin \left(\frac{\pi}{5}\right)} \sin \left(2 \times \frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \)

\( \rm p = \frac{1}{2 \sin \left(\frac{\pi}{5}\right)} \sin \left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \)

Now, we have \( \sin \left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \). We can apply the same identity again by multiplying and dividing by 2:

\( \rm p = \frac{1}{2 \sin \left(\frac{\pi}{5}\right)} \times \frac{1}{2} \left(2 \sin \left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\right) \)

\( \rm p = \frac{1}{4 \sin \left(\frac{\pi}{5}\right)} \sin \left(2 \times \frac{2 \pi}{5}\right) \)

\( \rm p = \frac{1}{4 \sin \left(\frac{\pi}{5}\right)} \sin \left(\frac{4 \pi}{5}\right) \)

We know that \( \sin(\pi - x) = \sin x \). So, \( \sin \left(\frac{4 \pi}{5}\right) = \sin \left(\pi - \frac{\pi}{5}\right) = \sin \left(\frac{\pi}{5}\right) \).

Substitute this into the expression for \(p\):

\( \rm p = \frac{1}{4 \sin \left(\frac{\pi}{5}\right)} \sin \left(\frac{\pi}{5}\right) \)

Assuming \( \sin \left(\frac{\pi}{5}\right) \neq 0 \), which is true since \(0 < \frac{\pi}{5} < \pi\), we can cancel the \( \sin \left(\frac{\pi}{5}\right) \) terms:

\( \rm p = \frac{1}{4} \)

Evaluating the Value of q

Next, let's evaluate the expression for \(q\):

\( \rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right) \)

We can simplify the angles using properties of cosine:

• \( \cos(\pi - x) = -\cos x \)
• \( \cos(2\pi - x) = \cos x \)

For the first term:

\( \cos \left(\frac{4 \pi}{5}\right) = \cos \left(\pi - \frac{\pi}{5}\right) = -\cos \left(\frac{\pi}{5}\right) \)

For the second term:

\( \cos \left(\frac{8 \pi}{5}\right) = \cos \left(2\pi - \frac{2 \pi}{5}\right) = \cos \left(\frac{2 \pi}{5}\right) \)

Substitute these simplified terms back into the expression for \(q\):

\( \rm q = \left(-\cos \left(\frac{\pi}{5}\right)\right) \left(\cos \left(\frac{2 \pi}{5}\right)\right) \)

\( \rm q = -\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \)

Notice that the expression \( \cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \) is exactly the expression for \(p\).

So, \( \rm q = -p \).

Since we found \( \rm p = \frac{1}{4} \), it follows that \( \rm q = -\frac{1}{4} \).

Calculating the Value of p + q

Now we need to find the sum \(p + q\):

\( \rm p + q = \frac{1}{4} + \left(-\frac{1}{4}\right) \)

\( \rm p + q = \frac{1}{4} - \frac{1}{4} \)

\( \rm p + q = 0 \)

The value of \(p + q\) is 0.

Revision Table: Trigonometric Angles and Identities

Additional Information: Key Trigonometric Concepts

This problem utilizes several fundamental concepts in trigonometry, essential for evaluating expressions involving angles and trigonometric functions.

• Product-to-Sum Identities: While we used a rearrangement of the double angle identity \(2 \sin A \cos A = \sin 2A\), related identities like \(2 \cos A \cos B = \cos(A-B) + \cos(A+B)\) exist and are useful for similar product evaluations.
• Angle Periodicity and Symmetry: Understanding the periodic nature of sine and cosine (period \(2\pi\)) and their symmetry properties around \(\pi\) and \(2\pi\) is crucial. For example, \( \cos(\theta + 2\pi k) = \cos \theta \) for any integer \(k\), and \( \cos(-\theta) = \cos \theta \). Also, \( \cos(\pi - \theta) = -\cos \theta \) and \( \sin(\pi - \theta) = \sin \theta \). These properties helped simplify \( \cos\left(\frac{4 \pi}{5}\right) \) and \( \cos\left(\frac{8 \pi}{5}\right) \).
• Special Angles: Although \(\frac{\pi}{5}\) and \(\frac{2\pi}{5}\) are not standard "special" angles like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), their trigonometric values can be derived (related to the golden ratio), but that was not necessary for this specific problem as terms cancelled out.
• Unit Circle: Visualizing angles on the unit circle helps understand the sign and value of trigonometric functions for angles outside the first quadrant. For instance, \(\frac{4\pi}{5}\) is in the second quadrant (where cosine is negative), and \(\frac{8\pi}{5}\) is in the fourth quadrant (where cosine is positive).

Mastering these concepts is key to solving a wide range of trigonometric problems.