Evaluate trigonometric expression

In summary, a trigonometric expression is a mathematical expression that involves trigonometric functions and is used to represent relationships between angles and sides of a triangle. To evaluate a trigonometric expression, one should substitute given values and use the correct order of operations and simplification. Common trigonometric identities used for evaluation include the Pythagorean identities, double angle identities, and sum and difference identities. Tips for simplifying trigonometric expressions include factoring, using identities, and converting functions. In real life, trigonometric expressions are used in fields such as engineering, physics, and astronomy for problem-solving and in navigation, surveying, and architecture for calculating distances and angles.
  • #1
anemone
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Without the help of calculator, evaluate $\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\cos \dfrac{4\pi}{7}$.
 
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  • #2
My solution:

Using the fact that [tex]\displaystyle \begin{align*} \sin{ (2X)} \equiv 2\sin{(X)}\cos{(X)} \end{align*}[/tex] we can see that [tex]\displaystyle \begin{align*} \cos{(X)} \equiv \frac{\sin{(2X)}}{2\sin{(X)}} \end{align*}[/tex]. From here

[tex]\displaystyle \begin{align*} \cos{(x)} &\equiv \frac{\sin{(2x)}}{2\sin{(x)}} \\ \cos{(2x)} &\equiv \frac{\sin{(4x)}}{2\sin{(2x)}} \\ \cos{(4x)} &\equiv \frac{\sin{(8x)}}{2\sin{(4x)}} \\ \vdots \\ \cos{\left( 2^j x \right) } &\equiv \frac{\sin{ \left( 2^{j + 1}x \right) }}{2\sin{ \left( 2^j x \right) }} \end{align*}[/tex]

and so

[tex]\displaystyle \begin{align*} \cos{(x)} \cdot \cos{(2x)} \cdot \cos{(4x)} \cdot \dots \cdot \cos{ \left( 2^j x \right) } &\equiv \frac{\sin{(2x)}}{2\sin{(x)}} \cdot \frac{\sin{(4x)}}{2\sin{(2x)}} \cdot \frac{\sin{(8x)}}{2\sin{(4x)}} \cdot \dots \cdot \frac{\sin{ \left( 2^{j + 1}x \right) }}{2 \sin{ \left( 2^j x \right) }} \\ &\equiv \frac{ \sin{ \left( 2^{j + 1}x \right) } }{ 2^{j + 1} \sin{(x)} } \end{align*}[/tex]

thereby giving the identity [tex]\displaystyle \begin{align*} \prod_{j = 0}^{k - 1}{\cos{\left( 2^j x \right) }} \equiv \frac{\sin{ \left( 2^k x \right) }}{2^k \sin{(x)}} \end{align*}[/tex], and setting [tex]\displaystyle \begin{align*} x = \frac{\pi}{7} \end{align*}[/tex] we have

[tex]\displaystyle \begin{align*} \cos{ \left( \frac{\pi}{7} \right) } \cos{ \left( \frac{2\pi}{7} \right) } \cos{ \left( \frac{4\pi}{7} \right) } &= \prod _{j = 0}^2{\cos{ \left( 2^j \cdot \frac{\pi}{7} \right) } } \\ &= \frac{ \sin{ \left( 2^3 \cdot \frac{\pi}{7} \right) }}{ 2^3 \sin{ \left( \frac{\pi}{7} \right) } } \\ &= \frac{\sin{\left( \frac{8\pi}{7} \right) }}{8\sin{ \left( \frac{\pi}{7} \right) }} \\ &= \frac{\sin{ \left( \pi + \frac{\pi}{7} \right) }}{8\sin{ \left( \frac{\pi}{7} \right) }} \\ &= \frac{-\sin{\left( \frac{\pi}{7} \right) }}{8\sin{ \left( \frac{\pi}{7} \right) }} \\ &= -\frac{1}{8} \end{align*}[/tex]
 
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  • #3
multiply by 8 $\sin(\pi/7) t$o get
$8\sin (\pi/7) cos (\pi/7)\ cos ( 2\pi/7) \cos(4\pi/7)$
=$4\ sin (2\pi/7) \cos ( 2\pi/7) \cos(4\pi/7)$
=$2 \sin (4\pi/7) \cos(4\pi/7)$
=$ \sin (8\pi/7)$
=$ -\sin (\pi/7) $
hence $ \cos (\pi/7) \cos ( 2\pi/7) \cos(4\pi/7)= -1/8$
 
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  • #4
anemone said:
Without the help of calculator, evaluate $\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\cos \dfrac{4\pi}{7}$.

Using the substitution $\cos x = \frac 1 2 \left(e^{ix} + e^{-ix}\right)$, we find:

$$\cos \frac{\pi}{7}\cos \frac{2\pi}{7}\cos \frac{4\pi}{7}
=\frac 1 8 \left(e^{i\pi/7} + e^{-i\pi/7}\right) \left(e^{i 2\pi/7} + e^{-i 2\pi/7}\right)
\left(e^{i 4\pi/7} + e^{-i 4\pi/7}\right)$$

Define $z=e^{i\pi/7}$ and this yields:

$$\frac 1 8 (z+z^{-1})(z^2+z^{-2})(z^4+z^{-4})
=\frac 1 8 (z^{-7} + z^{-5} + z^{-3} + z^{-1} + z^{1} + z^{3} + z^{5} + z^{7})$$
Using the formula for a geometric series, this rolls up into:
$$\frac 1 8 z^{-7} \frac{1-(z^2)^8}{1-z^2}$$

From the definition of $z$ it follows that $z^7 = -1$.
Consequently the expression simplifies to just \(\displaystyle -\frac 1 8\).

In other words:
$$\cos \frac{\pi}{7}\cos \frac{2\pi}{7}\cos \frac{4\pi}{7} = -\frac 1 8$$
 
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  • #5
Hey Prove It, kaliprasad and I like Serena,

Thank you so much for participating! The solutions provided by the three of you suggest that there are many ways to tackle a math problem, and this is especially true in trigonometric problem!:)

Another solution that I saw somewhere which I think is good to share it here:

$\begin{align*}\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\cos \dfrac{4\pi}{7}&=\dfrac{\left(2\sin \dfrac{\pi}{7}\cos \dfrac{\pi}{7} \right) \left(2\sin \dfrac{2\pi}{7}\cos \dfrac{2\pi}{7} \right) \left(2\sin \dfrac{4\pi}{7}\cos \dfrac{4\pi}{7} \right)}{8\sin \dfrac{\pi}{7}\sin \dfrac{2\pi}{7}\sin \dfrac{4\pi}{7}}\\&=\dfrac{\sin \dfrac{2 \pi}{7}\sin \dfrac{4\pi}{7}\sin \dfrac{8\pi}{7}}{8\sin \dfrac{\pi}{7}\sin \dfrac{2\pi}{7}\sin \dfrac{4\pi}{7}}\\&=\dfrac{\sin \left(\pi+\dfrac{\pi}{7} \right)}{8\sin \dfrac{\pi}{7}}\\&=-\dfrac{1}{8}\end{align*}$
 
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FAQ: Evaluate trigonometric expression

What is a trigonometric expression?

A trigonometric expression is a mathematical expression that involves trigonometric functions, such as sine, cosine, and tangent, as well as variables and constants. It is used to represent relationships between angles and sides of a triangle.

How do you evaluate a trigonometric expression?

To evaluate a trigonometric expression, you need to substitute the given values for the variables and then use the trigonometric functions to solve for the resulting numerical expression. Make sure to use the correct order of operations and to simplify the expression as much as possible.

What are the common trigonometric identities used to evaluate expressions?

Some common trigonometric identities used to evaluate expressions include the Pythagorean identities (sin²x + cos²x = 1, tan²x + 1 = sec²x, 1 + cot²x = csc²x), the double angle identities (sin2x = 2sinx cosx, cos2x = cos²x - sin²x), and the sum and difference identities (sin(x ± y) = sinx cosy ± cosx siny, cos(x ± y) = cosx cosy ∓ sinx siny).

What are some tips for simplifying trigonometric expressions?

Some tips for simplifying trigonometric expressions include factoring out common factors, using trigonometric identities, converting all trigonometric functions to sine and cosine, and substituting values for trigonometric functions using the unit circle.

How are trigonometric expressions used in real life?

Trigonometric expressions are used in various fields such as engineering, physics, and astronomy to solve problems involving angles and distances. They are also used in navigation, surveying, and architecture to calculate distances and angles between objects.

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