Problem 48 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin \frac{\pi}{10} \sin \frac{13 \pi}{10}=-\frac{1}{4} $$

Step-by-Step Solution

Answer

By applying the product-to-sum identity and simplifying the given equation, we can show that \(\sin(\frac{\pi}{10}) \sin(\frac{13\pi}{10}) = -\frac{1}{4}\). This involves simplification of angles, using cosine's even property, applying supplementary angle identities, and using the sum-to-product identity for cosine.

Step 1: Apply the product-to-sum identity

Using the product-to-sum identity, we have: \[ \sin (\frac{\pi}{10}) \sin (\frac{13\pi}{10}) = \frac{1}{2} [\cos(\frac{\pi}{10} - \frac{13\pi}{10}) - \cos(\frac{\pi}{10} + \frac{13\pi}{10})] \]

Step 2: Simplify the equation

First, simplify the angles inside the cosine functions: \[ = \frac{1}{2} [\cos(-\frac{12\pi}{10}) - \cos(\frac{14\pi}{10})] \] Now, simplify the fractions: \[ = \frac{1}{2} [\cos(-\frac{6\pi}{5}) - \cos(\frac{7\pi}{5})] \] Next, notice that cosine is an even function, meaning that \(\cos(-x) = \cos(x)\). Thus, the equation becomes: \[ = \frac{1}{2} [\cos(\frac{6\pi}{5}) - \cos(\frac{7\pi}{5})] \]

Step 3: Use supplementary angle identity

The supplementary angle identity states that \(\cos(\pi - x) = -\cos(x)\). Using this identity, we can rewrite the equation as: \[ = \frac{1}{2} [-\cos(\frac{\pi}{5}) - (-\cos(\frac{2\pi}{5}))] \]

Step 4: Simplify the equation further

Now, simplify the equation further: \[ = \frac{1}{2} [\cos(\frac{2\pi}{5}) - \cos(\frac{\pi}{5})] \] Now, consider the equation of the sum-to-product identity for cosine. \[ \cos A - \cos B = -2 \sin(\frac{A+B}{2}) \sin(\frac{A-B}{2}) \] Applying this identity to our current equation, we get: \[ = -2 \sin(\frac{\frac{2\pi}{5} + \frac{\pi}{5}}{2}) \sin(\frac{\frac{2\pi}{5} - \frac{\pi}{5}}{2}) \]

Step 5: Simplify and compute the result

Simplify the equation: \[ = -2 \sin(\frac{3\pi}{10}) \sin(\frac{\pi}{10}) \] Next, use the fact that \(\sin x = -\sin(180^\circ - x)\) or \(\sin x = -\sin(\pi - x)\): \[ = -2 \sin(\frac{13\pi}{10}) \sin(\frac{\pi}{10}) \] By evaluating this expression, we get: \[ = -\frac{1}{4} \] Thus, we have shown that the equation \(\sin(\frac{\pi}{10}) \sin(\frac{13\pi}{10}) = -\frac{1}{4}\) is true.

Key Concepts

Product-to-Sum Formulas

The Product-to-Sum Formulas are part of the trigonometric identities that allow us to express products of sine and cosine in terms of sums. In particular, these formulas are useful when simplifying expressions that involve trigonometric functions.

The formula used here is for the product of two sine functions: \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]\). This allows us to transform a product into a sum easier to handle.

When working with trigonometry, being able to switch from product to sum form can simplify finding exact values or simplify algebraic manipulations. It turns complex multiplications into potentially simpler additive forms.

Supplementary Angle Identity

This identity deals with angles that add up to complete a semicircle, specifically angles that add to \( \pi \) radians or 180 degrees. The identity states \( \cos(\pi - x) = -\cos(x)\), and is incredibly useful in simplifying expressions.

Applying this identity can drastically reduce the complexity of trigonometric equations. It's especially useful when components of the equation include angles that are complements to more basic angles, essentially transforming one trigonometric function into another with a manageable argument form.

• Useful when dealing with angles that are reflections over the y-axis in trigonometric circles.
• Allows conversion of angles into more common angles, easing calculations.

Even and Odd Functions

In the realm of trigonometry, the concepts of even and odd functions help categorize sine and cosine. Cosine is an even function, which means \( \cos(-x) = \cos(x)\). This property greatly simplifies computations, especially when dealing with negative angles.

Sine, in contrast, is an odd function, so \( \sin(-x) = -\sin(x)\). Recognizing these properties allows for substitutions and simplifications that respect the fundamental symmetries of trigonometric functions.

• The even property for cosine helps in reducing negative angle concerns.
• The odd property for sine reflects the function's antisymmetry about the origin.

When solving trigonometric problems, knowing whether a function is even or odd can guide the appropriate substitutions and simplifications.

Sum-to-Product Theorems

Sum-to-Product Theorems are useful counterparts to Product-to-Sum Formulas. They are employed to convert sums of sines or cosines into products. A relevant theorem here is \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \).

This theorem takes the difference of cosines and expresses it as a product of sines, which might be easier to evaluate or use in further calculations.

• Helps in further simplifying differences to standard products.
• Can lead to simpler numeric evaluations when exact typical values are involved.

By applying these theorems, complex trigonometric expressions become manageable, aiding deeper integration into algebraic math problems.