Problem 148 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin 4 A=4 \sin A \cos ^{3} A-4 \cos A \sin ^{3} A $$

Step-by-Step Solution

Answer

Yes, the given equation \(\sin 4A = 4 \sin A \cos ^{3} A - 4 \cos A \sin ^{3} A\) holds true.

Step 1: Express the left side of the equation in terms of double angles

We start by expressing \(\sin 4A\) in terms of double angles. \(\sin\) of double of any angle can be expressed as \(2 \sin x \cos x\). Hence, \(\sin 4A = 2 \sin 2A \cos 2A\).

Step 2: Expand \(\sin 2A\) and \(\cos 2A\)

We can now express \(\sin 2A\) and \(\cos 2A\) in terms of \(A\). Use the identities: \(\sin 2A = 2 \sin A \cos A\) and \(\cos 2A = \cos^2 A - \sin^2 A\). This gives \(\sin 4A = 2(2 \sin A \cos A)[\cos^2 A - \sin^2 A]\).

Step 3: Distribute \(2 \sin A \cos A\) into \([\cos^2 A - \sin^2 A]\)

We distribute \(2 \sin A \cos A\) into the terms inside the brackets: \[\sin 4A = 4 \sin A \cos ^{3} A - 4 \cos A \sin ^{3} A\]

Step 4: Comparison of both sides of the equation

We can see that the expanded form of \(\sin 4A\) matches with the right hand side of the equation, confirming that the equation holds true.

Key Concepts

Double Angle Formulas

The double angle formulas are a cornerstone of trigonometry, enabling the simplification and calculation of trigonometric functions of twice a certain angle in terms of the functions of the original angle. The fundamental double angle formulas include those for sine, cosine, and tangent:

• For sine, the formula is \[\begin{equation}\sin(2x) = 2\sin(x)\cos(x)\end{equation}\]
• For cosine, there are two equivalent formulas:
  • \[\begin{equation}\cos(2x) = \cos^2(x) - \sin^2(x)\end{equation}\]
  • \[\begin{equation}\cos(2x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)\end{equation}\]
• And for tangent:\[\begin{equation}\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\end{equation}\]

The exercise showcases the application of these formulas by expanding \( \sin 4A \) using the double angle formula for sine. We begin by considering \( \sin 4A \) as \( \sin (2 \cdot 2A) \), thus expressing it in the double angle form \( 2 \sin 2A \cos 2A \). Learning to manipulate these formulas allows students to solve a variety of trigonometric equations with ease.

Trigonometric Functions

Trigonometric functions are the relations between the angles and sides of a triangle, specifically a right-angled triangle. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). They are fundamental to the study of periodic phenomena, such as sound and light waves.

• \[\begin{equation}\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\end{equation}\]
• \[\begin{equation}\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\end{equation}\]
• \[\begin{equation}\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\end{equation}\]

Each function has an associated reciprocal function: cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). Understanding how to evaluate these functions for various angles and how to use them in conjunction with trigonometric identities is vital for solving trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations oftentimes requires identifying patterns and implementing trigonometric identities. To solve these equations effectively, certain strategies are essential:

• Manipulate the equation to express all terms in terms of a single trigonometric function, if possible.
• Use trigonometric identities to simplify or rewrite expressions.
• Isolate the trigonometric function and find its value for given intervals
• Carefully consider the domain of the function and the possible angles that satisfy the equation in that domain.

Applying these strategies will enable students to solve complex problems with relative ease. For instance, in the provided example, we expressed \( \sin 4A \) in terms of \( \sin A \) and \( \cos A \) by using the double angle formulas, which then allowed us to distribute and simplify the terms to arrive at the original equation. Understanding these techniques is crucial for students aiming to excel in trigonometry and its applications.