Problem 203 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ x=4 \tan ^{2} \frac{t}{2}, y=a \sin t+b \cos t $$

Step-by-Step Solution

Answer

The Cartesian equation relating x and y for the given parametric equations is: \(y = a \sqrt{\frac{x}{1 + \frac{x}{4}}} + b (1 - \frac{x}{4 + x})\).

Step 1: Analyzing the Parametric Equations

The given parametric equations are: \(x = 4 \tan^2{\frac{t}{2}}\) \(y = a \sin{t} + b \cos{t}\) Both equations are written in terms of the parameter t, which may represent time or another variable. The purpose of these parametric equations is to describe the x and y coordinates at different values of t.

Step 2: Eliminate the parameter t

To eliminate the parameter t, we can use one of the trigonometric identities. In this case, we can use the Pythagorean identity, which states: \(\sin^2{\theta} + \cos^2{\theta} = 1\) We know that \(x = 4 \tan^2{\frac{t}{2}}\), so we can rewrite this equation using the relationship between tan and sin, cos: \(x = 4 \frac{\sin^2{\frac{t}{2}}}{\cos^2{\frac{t}{2}}}\) Now we need to find a relationship between \(\sin{t}\) and \(\cos{t}\) using \(y = a \sin{t} + b \cos{t}\). We can get \(\sin{t}\) and \(\cos{t}\) in terms of x by using the double angle identities: \(\sin{t} = 2 \sin{\frac{t}{2}} \cos{\frac{t}{2}}\) \(\cos{t} = \cos^2{\frac{t}{2}} - \sin^2{\frac{t}{2}}\) Now we can plug these expressions into the equation for y, and we will get an equation relating y to x: \(y = a (2 \sin{\frac{t}{2}} \cos{\frac{t}{2}}) + b (\cos^2{\frac{t}{2}} - \sin^2{\frac{t}{2}})\) Now, we can use the equation \(x = 4 \frac{\sin^2{\frac{t}{2}}}{\cos^2{\frac{t}{2}}}\) to find \(\sin^2{\frac{t}{2}}\) and \(\cos^2{\frac{t}{2}}\) in terms of x, then substitute these expressions into the equation for y. \(\sin^2{\frac{t}{2}} = \frac{x}{4}\cos^2{\frac{t}{2}}\) \(\cos^2{\frac{t}{2}} = \frac{1}{1 + \frac{x}{4}}\) Plug the expressions for \(\sin^2{\frac{t}{2}}\) and \(\cos^2{\frac{t}{2}}\) into the equation for y:

Step 3: Simplify the Equation

Replace \(\sin^2{\frac{t}{2}}\) and \(\cos^2{\frac{t}{2}}\) with their expressions in terms of x: \(y = a (2 \sqrt{\frac{x}{4}}\sqrt{\frac{1}{1 + \frac{x}{4}}}) \sqrt{\frac{1}{1 + \frac{x}{4}}} + b (\frac{1}{1 + \frac{x}{4}} - \frac{x}{4}\frac{1}{1 + \frac{x}{4}})\) Now, simplify the expression: \(y = a \sqrt{\frac{x}{1 + \frac{x}{4}}} + b (1 - \frac{x}{4 + x})\)

Step 4: Final Result

We have successfully eliminated the parameter t from the parametric equations and found an equation relating x and y: \(y = a \sqrt{\frac{x}{1 + \frac{x}{4}}} + b (1 - \frac{x}{4 + x})\)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics, particularly when working with trigonometric functions such as sine, cosine, and tangent. These identities allow us to simplify complex trigonometric expressions and find relationships between them.

Some of the most common trigonometric identities include:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal Identities: \( \sec\theta = \frac{1}{\cos\theta} \), \( \csc\theta = \frac{1}{\sin\theta} \), \( \cot\theta = \frac{1}{\tan\theta} \)
• Double Angle Formulas: \( \sin{2\theta} = 2\sin\theta\cos\theta \), \( \cos{2\theta} = \cos^2\theta - \sin^2\theta \)

In the given exercise, we used the double angle formulas to express \( \sin{t} \) and \( \cos{t} \) in terms of \( \sin{\frac{t}{2}}\) and \(\cos{\frac{t}{2}}\). This helped us eliminate the parameter and find a relationship between \(x\) and \(y\).

Pythagorean Identity

The Pythagorean Identity is a staple in trigonometry that stems from the Pythagorean theorem. It is crucial for relating the sine and cosine functions of the same angle.

The identity is expressed as:

• \( \sin^2\theta + \cos^2\theta = 1 \)

This identity implies that no matter the value of \(\theta\), the combined squares of sine and cosine will always equal one. This is particularly useful for simplifying expressions and eliminating parameters.

In the step-by-step solution, we rewrote \(x = 4 \tan^2{\frac{t}{2}}\) using the Pythagorean identity after expressing tangent in terms of sine and cosine using \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). This approach allowed us to interrelate \(x\) and \(y\) through trigonometric identities.

Eliminating Parameters

Eliminating parameters in parametric equations is all about expressing one variable solely concerning one another, without involving the parameter, usually \( t \).

Here are some typical methods used for eliminating parameters:

• Using trigonometric identities
• Solving one equation for the parameter and substituting it into the other
• Employing inverse trigonometric functions

In our solution, we used the Pythagorean identity and trigonometric conversion formulas such as \(\sin t = 2 \sin \frac{t}{2} \cos \frac{t}{2}\) to express both \( \sin t \) and \( \cos t \) in terms of \( x \). Subsequently, we eliminated the parameter \( t \) by substituting these expressions into the equation for \( y \).

Double Angle Formulas

The double angle formulas are essential mathematical tools that simplify the process of expressing trigonometric functions of twice an angle in terms of single angles.

These formulas are expressed as follows:

• \( \sin{2\theta} = 2\sin\theta\cos\theta \)
• \( \cos{2\theta} = \cos^2\theta - \sin^2\theta \) or \( 1 - 2\sin^2\theta \) or \( 2\cos^2\theta - 1 \)

In the problem scenario, the double angle formulas provided a streamlined way to articulate \( \sin t \) and \( \cos t \) using half-angle specifics \( \sin\frac{t}{2} \) and \( \cos\frac{t}{2} \). This strategy was vital in transitioning from parameter-dependent expressions to a formula linking \(x\) and \(y\) directly. Utilizing these formulas simplifies complex trigonometric computations and enables clear solutions in parametric equations.