Consider the following for the next items that follow: Let \(\rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \) and \(\rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)\).
What is the value of pq?
Evaluating Trigonometric Products and Their Value
The problem asks for the value of the product of two given expressions, \(\rm p\) and \(\rm q\), which involve products of cosine functions with specific angles.
The expressions are given as:
- \(\rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\)
- \(\rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)\)
We need to find the value of \(\rm pq\).
Step 1: Evaluate the value of p
The expression for \(\rm p\) is \(\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\). This is a product of two cosine terms where the angles are in a geometric progression (\(\frac{\pi}{5}\) and \(\frac{2\pi}{5}\)). A common technique to evaluate such products is to multiply by the sine of the smallest angle and use the double angle identity \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\).
Let's multiply \(\rm p\) by \(2 \sin\left(\frac{\pi}{5}\right)\):
\(\displaystyle 2 \sin\left(\frac{\pi}{5}\right) \rm p = 2 \sin\left(\frac{\pi}{5}\right) \cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\)
Using the double angle identity \(2 \sin(\theta) \cos(\theta) = \sin(2\theta)\) for \(\theta = \frac{\pi}{5}\):
\(\displaystyle 2 \sin\left(\frac{\pi}{5}\right) \rm p = \sin\left(2 \times \frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\)
\(\displaystyle 2 \sin\left(\frac{\pi}{5}\right) \rm p = \sin\left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\)
Now, multiply both sides by 2 again:
\(\displaystyle 4 \sin\left(\frac{\pi}{5}\right) \rm p = 2 \sin\left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\)
Using the double angle identity again for \(\theta = \frac{2\pi}{5}\):
\(\displaystyle 4 \sin\left(\frac{\pi}{5}\right) \rm p = \sin\left(2 \times \frac{2 \pi}{5}\right)\)
\(\displaystyle 4 \sin\left(\frac{\pi}{5}\right) \rm p = \sin\left(\frac{4 \pi}{5}\right)\)
Now, we can express \(\sin\left(\frac{4 \pi}{5}\right)\) in terms of \(\sin\left(\frac{\pi}{5}\right)\) using the identity \(\sin(\pi - \theta) = \sin(\theta)\). For \(\theta = \frac{\pi}{5}\), we have \(\sin\left(\pi - \frac{\pi}{5}\right) = \sin\left(\frac{4 \pi}{5}\right)\). So, \(\sin\left(\frac{4 \pi}{5}\right) = \sin\left(\frac{\pi}{5}\right)\).
Substituting this back:
\(\displaystyle 4 \sin\left(\frac{\pi}{5}\right) \rm p = \sin\left(\frac{\pi}{5}\right)\)
Since \(\frac{\pi}{5}\) is in the first quadrant, \(\sin\left(\frac{\pi}{5}\right) \ne 0\), so we can divide both sides by \(\sin\left(\frac{\pi}{5}\right)\):
\(\displaystyle 4 \rm p = 1\)
\(\displaystyle \rm p = \frac{1}{4}\)
Step 2: Evaluate the value of q
The expression for \(\rm q\) is \(\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)\).
Let's simplify the arguments of the cosine functions using angle properties:
- \(\displaystyle \cos\left(\frac{4 \pi}{5}\right)\). We can use the identity \(\cos(\pi - \theta) = -\cos(\theta)\). For \(\theta = \frac{\pi}{5}\), we get \(\cos\left(\pi - \frac{\pi}{5}\right) = \cos\left(\frac{4 \pi}{5}\right)\). So, \(\cos\left(\frac{4 \pi}{5}\right) = -\cos\left(\frac{\pi}{5}\right)\).
- \(\displaystyle \cos\left(\frac{8 \pi}{5}\right)\). We can use the identity \(\cos(2\pi - \theta) = \cos(\theta)\). For \(\theta = \frac{2\pi}{5}\), we get \(\cos\left(2\pi - \frac{2 \pi}{5}\right) = \cos\left(\frac{10\pi - 2\pi}{5}\right) = \cos\left(\frac{8 \pi}{5}\right)\). So, \(\cos\left(\frac{8 \pi}{5}\right) = \cos\left(\frac{2 \pi}{5}\right)\).
Substitute these simplified terms back into the expression for \(\rm q\):
\(\displaystyle \rm q = \left(-\cos\left(\frac{\pi}{5}\right)\right) \left(\cos\left(\frac{2 \pi}{5}\right)\right)\)
\(\displaystyle \rm q = - \left(\cos\left(\frac{\pi}{5}\right) \cos\left(\frac{2 \pi}{5}\right)\right)\)
Notice that the term inside the parentheses is exactly the expression for \(\rm p\) that we evaluated in Step 1. We found that \(\rm p = \cos\left(\frac{\pi}{5}\right) \cos\left(\frac{2 \pi}{5}\right) = \frac{1}{4}\).
Substituting the value of \(\rm p\) into the expression for \(\rm q\):
\(\displaystyle \rm q = - \left(\frac{1}{4}\right)\)
\(\displaystyle \rm q = -\frac{1}{4}\)
Step 3: Calculate the value of pq
Now we need to find the product of \(\rm p\) and \(\rm q\).
\(\displaystyle \rm pq = \rm p \times \rm q\)
Substitute the values we found for \(\rm p\) and \(\rm q\):
\(\displaystyle \rm pq = \left(\frac{1}{4}\right) \times \left(-\frac{1}{4}\right)\)
\(\displaystyle \rm pq = -\frac{1 \times 1}{4 \times 4}\)
\(\displaystyle \rm pq = -\frac{1}{16}\)
Thus, the value of \(\rm pq\) is \(\displaystyle -\frac{1}{16}\).
Result Summary
| Expression | Evaluated Value |
|---|---|
| \(\displaystyle \rm p = \cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)\) | \(\displaystyle \frac{1}{4}\) |
| \(\displaystyle \rm q = \cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)\) | \(\displaystyle -\frac{1}{4}\) |
| \(\rm pq\) | \(\displaystyle -\frac{1}{16}\) |
The final value of \(\rm pq\) is \(\displaystyle -\frac{1}{16}\).
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\) | Double angle identity for sine. Useful for simplifying products of sine and cosine. |
| \(\sin(\pi - \theta) = \sin(\theta)\) | Sine of a supplementary angle. |
| \(\cos(\pi - \theta) = -\cos(\theta)\) | Cosine of a supplementary angle. |
| \(\cos(2\pi - \theta) = \cos(\theta)\) | Cosine property for angles differing by \(2\pi\). |
Additional Information: Evaluating Cosine Products like \(\cos(\pi/n) \cos(2\pi/n) \cdots \cos((n-1)\pi/n)\)
The product \(\cos(\pi/5) \cos(2\pi/5)\) is part of a more general type of product. Products of the form \(\cos(\theta) \cos(2\theta) \cos(4\theta) \cdots \cos(2^{k-1}\theta)\) can be simplified using repeated application of the double angle identity, typically by multiplying by \(2\sin(\theta)\).
For example, \(\cos(\theta) \cos(2\theta) = \frac{2\sin(\theta)\cos(\theta)\cos(2\theta)}{2\sin(\theta)} = \frac{\sin(2\theta)\cos(2\theta)}{2\sin(\theta)} = \frac{2\sin(2\theta)\cos(2\theta)}{4\sin(\theta)} = \frac{\sin(4\theta)}{4\sin(\theta)}\).
Applying this pattern, the product \(\cos(\theta) \cos(2\theta) \cdots \cos(2^{k-1}\theta) = \frac{\sin(2^k \theta)}{2^k \sin(\theta)}\).
In our case for \(\rm p\), we have \(\theta = \frac{\pi}{5}\) and the product is \(\cos(\frac{\pi}{5}) \cos(2 \times \frac{\pi}{5})\), which matches the form \(\cos(\theta) \cos(2\theta)\) with \(k=2\). So, using the formula: \(\rm p = \frac{\sin(2^2 \times \pi/5)}{2^2 \sin(\pi/5)} = \frac{\sin(4\pi/5)}{4\sin(\pi/5)}\). As shown earlier, \(\sin(4\pi/5) = \sin(\pi - \pi/5) = \sin(\pi/5)\). Thus, \(\rm p = \frac{\sin(\pi/5)}{4\sin(\pi/5)} = \frac{1}{4}\).
This general formula provides a quicker way to evaluate such specific trigonometric products.
Important Questions from Trigonometric Functions
If \(\tan \alpha=\frac{1}{7}\), \(\sin \beta=\frac{1}{\sqrt{10}}\); \(0<\alpha, \beta<\frac{\pi}{2}\), then what is the value of cos (α + 2β) ?
What is the period of the function?
What is the value of p + q?
What is pq equal to ?
What is the derivative of \({\tan ^{ - 1}}\left( {\frac{{\sqrt {1{\rm{\;}} + {{\rm{x}}^2}} - {\rm{\;}}1}}{{\rm{x}}}} \right)\) with respect to tan -1 x ?
