Problem 52 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } x=y \cos \frac{2 \pi}{3}=z \cos \frac{4 \pi}{3}, \text { then show that } x y+y z+z x=0 $$

Step-by-Step Solution

Answer

We are given \(x = y \cos{\frac{2 \pi}{3}}\) and \(x = z \cos{\frac{4 \pi}{3}}\). By solving for y and z, we get \(y = -2x\) and \(z = -2x\). Substituting these back into the expression \(xy + yz + zx\), we get \((-2x)(x) + (-2x)(-2x) + (-2x)(x) = -2x^2 + 4x^2 - 2x^2\), which simplifies to \(0\). Therefore, we have shown that \(xy + yz + zx = 0\).

Step 1: Express x, y, and z in terms of trigonometric functions

From the given equation, we have the following relationships: \(x = y \cos{\frac{2 \pi}{3}}\) \(x = z \cos{\frac{4 \pi}{3}}\)

Step 2: Solve for y and z in terms of x

Using the relationships found in Step 1, we can solve for y and z in terms of x as follows: \(y = \frac{x}{\cos{\frac{2 \pi}{3}}}\) \(z = \frac{x}{\cos{\frac{4 \pi}{3}}}\)

Step 3: Simplify trigonometric functions

Using the cosine properties, simplify \(\cos{\frac{2 \pi}{3}}\) and \(\cos{\frac{4 \pi}{3}}\): \(\cos{\frac{2 \pi}{3}} = -\frac{1}{2}\) \(\cos{\frac{4 \pi}{3}} = -\frac{1}{2}\)

Step 4: Calculate y and z in terms of x

Now that we've simplified the trigonometric functions, we can find the expressions for y and z in terms of x: \(y = \frac{x}{-\frac{1}{2}} = -2x \) \(z = \frac{x}{-\frac{1}{2}} = -2x \)

Step 5: Substitute y and z into the target expression

We want to show that \(xy + yz + zx = 0\). Substitute the expressions for y and z in terms of x from Step 4: \((-2x)(x) + (-2x)(-2x) + (-2x)(x) \)

Step 6: Simplify the expression

Simplifying the expression obtained in Step 5: \(-2x^2 + 4x^2 - 2x^2 \)

Step 7: Verify that the expression equals 0

Add up all the terms in the expression from Step 6: \(0 = -2x^2 + 4x^2 -2x^2\) We have successfully shown that \(xy + yz + zx = 0\).

Key Concepts

Cosine Angle

Understanding cosine angles is fundamental to solving trigonometric equations. In trigonometry, the cosine of an angle is a measure of the adjacent side's length divided by the hypotenuse in a right triangle.
When dealing with angles beyond the first quadrant, such as \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3}\), we must consider the unit circle. These angles are located in the second and third quadrants respectively.
The cosines of these specific angles, \( \cos\frac{2\pi}{3} \) and \( \cos\frac{4\pi}{3} \), both equal \(-\frac{1}{2}\). This simplification often eases the solving process of complex equations. Remembering these cosine values can greatly assist when solving problems with similar setups.

Trigonometric Expressions

Working with trigonometric expressions involves understanding how functions like sine and cosine can influence an equation.
In this exercise, we start by expressing the variable \( x \) in terms of other variables using trigonometric functions, which is a crucial step in simplifying expressions.
The equations \( x = y \cos\frac{2\pi}{3} \) and \( x = z \cos\frac{4\pi}{3} \) allow us to express \( y \) and \( z \) in relation to \( x \).
This conversion is essential because it provides a pathway to a more simplified expression that we can solve. Learning to manipulate and transform these expressions typically requires practice but is an invaluable skill in further trigonometric studies.

Problem Solving

Problem-solving with trigonometric identities involves systematic steps to reach a solution. Start by expressing all given requirements with the help of trigonometric properties.
Here, understanding cosine values helped simplify \( y \) and \( z \) in terms of \( x \), transforming a seemingly complex problem into simpler terms. 

• First, expressed \( y \) and \( z \) in terms of \( x \) using given cosine values.
• Then substituted these values back into the target expression \( xy + yz + zx = 0 \).
• Simplified the expression to verify the solution.

Approaching problems through structured steps not only organizes thinking but also enhances learning by confirming the correctness of each action performed. Continuous practice of these strategies increases confidence and efficiency in tackling trigonometric problems.