Problem 219 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{6 \pi}{15} \cos \frac{7 \pi}{15}=\frac{1}{2^{7}} $$

Step-by-Step Solution

Answer

The statement is correct. The product of the cosines is equal to \(\frac{1}{2^{7}}\).

Step 1: Pair Up Input Values

Split the product of cosine values into sets of pairs by combining cosine terms to get : \( cos(\frac{\pi}{15}) cos(\frac{7\pi}{15}) \), \( cos(\frac{2\pi}{15}) cos(\frac{6\pi}{15}) \) , \( cos(\frac{3\pi}{15}) cos(\frac{5\pi}{15}) \) and \( cos(\frac{4\pi}{15}) \).

Step 2: Apply Cosine Sum of Angles Identity

For each group, apply the identity \( cos(a) cos(b) = 1/2[cos(a+b)+cos(a-b)] \) to simplify the expression. You'll get: \( 1/2[ cos(\frac{8\pi}{15})+cos(\pi/15) ] \), \( 1/2[ cos(\frac{8\pi}{15})+cos(\pi/15) ] \) , \( 1/2[ cos(\frac{8\pi}{15})+cos(\pi/15) ] \) and \( cos(\frac{4\pi}{15}) \).

Step 3: Substitute cosine of sum formula and solve

The next step is to combine these pairs together and substitute the cosine sum formula. After this is done, the next step is to multiply the whole product together which will yield \( (1/2)^{3} cos(\frac{4\pi}{15}) \).

Step 4: Finally, Simplify

Since \$ cos(\frac{4\pi}{15}) \$ is equal to \$ cos(\pi/2 - \frac{\pi}{15}) \$ and \$ cos(\pi/2 - x) \$ is equal to \$ sin(x) \$, the solution is \( (1/2)^{3} sin(\frac{\pi}{15}) = (1/2)^{7} \), so \( sin(\frac{\pi}{15}) \$ must be equal to \$ (1/2)^{4} \)

Key Concepts

Trigonometric Ratios

Understanding trigonometric ratios is essential when dealing with angles and the corresponding ratios of the sides of right triangles. The most common trigonometric ratios are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively. These ratios help represent the relationships between the angles and sides in a triangle.
For instance:

• Sine (\( ext{sin} heta \)) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( ext{cos} heta \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan} heta \)) is the ratio of the opposite side to the adjacent side.

The cosine of an angle can sometimes be simplified by using identities or angle sum formulas, notably in exercises where products of cosine terms need to be converted using the Cosine Angle Sum formulas.

Cosine Angle Sum

The Cosine Angle Sum identity helps in expressing the cosine of the sum of two angles as a product of two trigonometric ratios of those angles. This identity is given by:

• \[\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\]

This formula can be rearranged to fit a product form for simplifying expressions containing terms like \(\cos(a) \cos(b)\). This is particularly useful in exercises where you need to simplify a product of multiple cosine terms. Utilizing this identity can help combine and reduce trigonometric terms into simpler forms, which can then be easily computed or used in further derivations.

Product to Sum Formulas

Product to Sum formulas are useful tools in trigonometry for converting the product of cosine or sine functions into a sum or difference of cosines or sines. This simplifies the process of solving equations or evaluating products. The relevant formula for cosine is:

• \[\cos(a) \cos(b) = \frac{1}{2} \big[ \cos(a+b) + \cos(a-b) \big]\]

For example, if you encounter a product such as \( \cos(\frac{\pi}{15}) \cos(\frac{7\pi}{15}) \), the formula converts it into a sum, making it easier to handle.

• Apply the formula to each pair in the product.
• Simplify the resulting expression, particularly focusing on any symmetries or known values of cosine for specific angles.

This transition from product to sum is especially valuable for computing products of multiple angles, thus aiding in reaching simplified forms or exact results.