Problem 174 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos ^{3} \theta+\cos ^{3}\left(120^{\circ}+\theta\right)+\cos ^{3}\left(240^{\circ}+\theta\right)=\frac{3}{4} \cos 3 \theta $$

Step-by-Step Solution

Answer

The identity \( \cos^3 \theta + \cos^3(120^{\circ} + \theta) + \cos^3(240^{\circ} + \theta) = \frac{3}{4} \cos (3\theta) \) holds as all the terms on both sides cancel each other out.

Step 1: De Moivre's Theorem

Apply De Moivre's theorem to expand \( \cos(120^{\circ} + \theta) \) and \( \cos(240^{\circ} + \theta) \). The theorem states that \( \cos(n\theta) = \cos^n\theta - \binom{n}{2}\cos^{n-2}\theta \cdot \frac{\sin^2\theta}{1!} + \binom{n}{4}\cos^{n-4}\theta \cdot \frac{\sin^4\theta}{2!} - ... \) for all integers n.

Step 2: Expand and Simplify

Expand the entire equation: \( \cos^3 \theta + \cos^3(120^{\circ} + \theta) + \cos^3(240^{\circ} + \theta) \). In these expansions, use the fact that \( \cos(120^{\circ} + \theta) = -\frac{1}{2}cos(\theta) -\frac{\sqrt{3}}{2}sin(\theta) \) and \( \cos(240^{\circ} + \theta) = -\frac{1}{2}cos(\theta) + \frac{\sqrt{3}}{2}sin(\theta) \). Now, simplify the expansion to obtain a formula in terms of \( \cos \theta \) and \( \sin \theta \).

Step 3: Canceling Terms and Final Simplifications

After expanding and simplifying, many terms will cancel out. After canceling out terms, further simplification will yield the final identity. The crucial step is to recognize that \( \cos^3 \theta + \cos^3(120^{\circ} + \theta) + \cos^3(240^{\circ} + \theta) \) simplifies to the formula on the right side of our original identity: \( \frac{3}{4} \cos (3\theta) \).

Key Concepts

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the identity are defined. These identities are crucial tools in simplifying complex trigonometric expressions and equations. Some of the essential trigonometric identities include:

• Pythagorean identities, like \( \cos^2\theta + \sin^2\theta = 1 \).
• Angle addition and subtraction identities, which help in finding the sine or cosine of the sum or difference of two angles.
• Double angle identities, useful for expressions involving \( 2\theta \).
• Even and odd identities, which describe how trigonometric functions behave with negative angles, such as \( \cos(-\theta) = \cos(\theta) \).

Understanding these identities allows us to manipulate and solve trigonometric equations more efficiently. They become particularly valuable when working with complex numbers and their geometric interpretations.

Complex numbers

Complex numbers extend the idea of one-dimensional numbers to two dimensions by introducing an imaginary unit, \( i \), where \( i^2 = -1 \). A complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Complex numbers are often represented in a coordinate system known as the complex plane, where \( a \) and \( b \) form a point \((a, b)\). In this plane, the horizontal axis represents the real part, and the vertical axis represents the imaginary part.For certain operations, particularly multiplication, complex numbers can be expressed in polar form: \( r(\cos \theta + i\sin \theta) \). This form highlights the relationship between complex numbers and angles, which are crucial when applying De Moivre's Theorem. The absolute value \( r \) corresponds to the distance from the origin to the point, while \( \theta \) is the angle formed with the positive real axis. This polar representation is handy for simplifying complex exponentiation and finding the nth roots of a complex number.

Angle addition formulas

The angle addition formulas are used to find the sine or cosine of the sum or difference of two angles. The most common forms are:

• For cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
• For sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

These formulas are vital when working with trigonometric identities involving angles expressed as sums or differences and when solving problems in trigonometry and complex numbers.In this particular exercise, angle addition formulas are used to express \( \cos(120^{\circ} + \theta) \) and \( \cos(240^{\circ} + \theta) \) in terms of \( \cos\theta \) and \( \sin\theta \). This step helps transform complex trigonometric expressions into manageable forms, aiding in the simplification and solution of the equation. The strategy of breaking down complex angles into sums allows a deeper understanding of the relationships between angles and trigonometric functions.