Problem 17 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(y=\cos \theta+i \sin \theta\), then the value of \(y+\frac{1}{y}\) is (a) \(2 \cos \theta\) (b) \(2 \sin \theta\) (c) \(2 \operatorname{cosec} \theta\) (d) \(2 \tan \theta\)

Step-by-Step Solution

Answer

The value of \( y + \frac{1}{y} \) is \( 2 \cos \theta \); option (a).

Step 1: Set Up the Expression for y

Given that \( y = \cos \theta + i \sin \theta \), we recognize this as similar to Euler's formula, where \( y = e^{i \theta} \).

Step 2: Find \(\frac{1}{y}\)

Since \( y = e^{i \theta} \), the reciprocal \( \frac{1}{y} \) is \( e^{-i \theta} \), which is equal to \( \cos \theta - i \sin \theta \).

Step 3: Calculate \( y + \frac{1}{y} \)

Substitute \( y = \cos \theta + i \sin \theta \) and \( \frac{1}{y} = \cos \theta - i \sin \theta \) into the expression \( y + \frac{1}{y} \).\[y + \frac{1}{y} = (\cos \theta + i \sin \theta) + (\cos \theta - i \sin \theta) = 2 \cos \theta.\]

Step 4: Select the Correct Option

Based on our calculations in Step 3, \( y + \frac{1}{y} = 2 \cos \theta \). Therefore, the correct answer is option (a) \( 2 \cos \theta \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They can be expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) representing the square root of \( -1 \). This special number \( i \) follows the fundamental identity \( i^2 = -1 \).
Complex numbers appear in various fields such as electrical engineering and quantum physics, providing a way to handle calculations involving two-dimensional quantities. When plotting on a complex plane, the real part is represented on the horizontal axis, and the imaginary part is represented on the vertical axis.

• The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \).
• Adding and subtracting complex numbers involves combining like parts: real with real, imaginary with imaginary.

When used in trigonometry, complex numbers allow the expression of functions like sine and cosine as part of exponential functions, which is particularly beneficial when using Euler's Formula.

Euler's Formula

Euler's Formula is a beautiful bridge between exponential functions and trigonometric functions. It states that for any real number \( \theta \), the complex exponential \( e^{i\theta} \) can be expressed as \( \cos \theta + i \sin \theta \).
This formula shows the deep connection between exponential growth and oscillation, allowing complex numbers to represent rotations and oscillations in a simple form. Because of Euler's Formula, any rotation in the plane can be represented as a multiplication by a complex number \( e^{i\theta} \):

• \( e^{0\cdot i} = 1 \) (no rotation)
• \( e^{\pi\cdot i} = -1 \) (half-turn or direct opposite)
• Adding the argument \( \theta \) results in rotating the point in the complex plane by \( \theta \).

A real-world application includes using this formula for solving differential equations and modeling wave functions in physics, making it essential in understanding the cyclical patterns.

Reciprocal Identities

Reciprocal identities are basic trigonometric identities involving the reciprocals of sine, cosine, and tangent functions. Each function has a reciprocal counterpart:

• The reciprocal of sine is cosecant, \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• The reciprocal of cosine is secant, \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• The reciprocal of tangent is cotangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

These identities are particularly useful in simplifying complex trigonometric expressions and solving trigonometric equations where direct calculation of the function is cumbersome. In our problem, recognizing the expressions like \( y = \cos \theta + i \sin \theta \) and its reciprocal could bridge the understanding of trigonometric forms with other mathematical fields like complex analysis.