Problem 152 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos ^{2} a+\cos ^{2}\left(a+120^{\circ}\right)+\cos ^{2}\left(a-120^{\circ}\right)=\frac{3}{2} $$

Step-by-Step Solution

Answer

The solutions for the trigonometric equation are \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

Step 1: Use the Trigonometric Identity

Expand each term using the cos identity \( \cos(120^o) = -\frac{1}{2} \). Thus, the equation becomes \( \cos^2 a + \cos^2(a - \frac{1}{2}) + \cos^2(a + \frac{1}{2}) = \frac{3}{2} \)

Step 2: Simplify the Equation

Substitute \( \cos^2 x = 1 - \sin^2 x \) into the equation and get three equations: \( \cos^2 a , \(1 - \sin^2(a - \frac{1}{2}), and \( 1 - \sin^2(a + \frac{1}{2}) = \frac{3}{2} \)

Step 3: Solve the Equation

Add up all three equations and solve. This gives the final solution as \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

Key Concepts

Cosine functions

When tackling problems involving cosine functions, it is crucial to understand their behavior and key properties.

• The cosine function, denoted as \( \cos(\theta) \), is one of the primary trigonometric functions. It represents the x-coordinate of a point on the unit circle that makes an angle \( \theta \) with the positive y-axis.
• Cosine is an even function, meaning \( \cos(-\theta) = \cos(\theta) \). This property can be very helpful in simplifying equations.
• The cosine function cycles every \( 360^\circ \) (or \( 2\pi \) radians). This means \( \cos(\theta + 360^\circ) = \cos(\theta) \).

Understanding these properties allows us to expand cosine-related expressions effectively. For instance, in the example given, the cosine function at an angle of \( 120^\circ \) is used. This involves using a trigonometric identity which leads to further simplification.

Trigonometric equations

Solving trigonometric equations requires a deep understanding of identities and transformations that can simplify the expressions.

• Trigonometric identities like \( \cos^2 x = 1 - \sin^2 x \) can convert between sine and cosine, helping to simplify complex equations.
• The given exercise illustrates the usage of trigonometric identities by substituting cosine with sine for simplification. This reduces the complexity of the problem, making it approachable through standard algebraic techniques.
• When solving these equations, look for patterns or known identities that convert the expression into a solvable form.

Always double-check each step to ensure no crucial transformation is missed. This discipline reaffirms understanding and accuracy when seeking solutions like those shown for cosine squared expressions.

Angle transformations

In trigonometry, angle transformations allow us to manipulate the angles in the functions for easier calculations.

• Common transformations involve using angles like \( a + 120^\circ \) or \( a - 120^\circ \), which may lead to known values or identities that simplify the computations.
• In our example, angles are adjusted by \( 120^\circ \) to utilize the identity: \( \cos(120^\circ) = -\frac{1}{2} \). This helps in rearranging and solving the equation.
• Understanding the implications of these transformations is essential to converting challenging expressions into easier ones. It also helps translate the trigonometric function into solvable elements of simpler functions.

By practicing these transformations, you'll start visualizing angles in different forms, revealing shortcuts and simplifications to solve many trigonometric problems efficiently.