Simplifying Trigonometric Expressions Using Identities

Trigonometry can often become complex when dealing with higher-order trigonometric functions like cos 3A and sin 3A. This article will guide you through simplifying the expression (frac{3cos A cos 3A}{3sin A - sin 3A}) using well-known trigonometric identities and compound angle formulas. We will explore the use of these identities and formulas to make the simplification process clearer and more understandable.

Introduction

In trigonometry, simplifying expressions is crucial for solving complex equations and understanding the relationships between various trigonometric functions. One such expression that can be simplified is (frac{3cos A cos 3A}{3sin A - sin 3A}). This article will demonstrate how to simplify this expression using the triple angle formulas and compound angle formulas.

Using Triple Angle Formulas

The triple angle formulas are useful in simplifying trigonometric expressions involving terms like cos 3A and sin 3A. These formulas are stated as:

(cos 3A 4cos^3 A - 3cos A) (sin 3A 3sin A - 4sin^3 A)

By applying these formulas, we can break down the original expression into simpler components and then simplify it.

First, let's substitute the triple angle formulas into the expression:

Numerator:

(3cos A cos 3A 3cos A (4cos^3 A - 3cos A) 4cos^3 A)

Denominator:

(3sin A - sin 3A 3sin A - (3sin A - 4sin^3 A) 4sin^3 A)

Now, we can rewrite the entire expression as:

(frac{3cos A cos 3A}{3sin A - sin 3A} frac{4cos^3 A}{4sin^3 A} frac{cos^3 A}{sin^3 A} cot^3 A)

Therefore, the simplified expression is:

(frac{3cos A cos 3A}{3sin A - sin 3A} cot^3 A)

Using Sum of Function Formulas

Alternatively, we can use the sum of function formulas for cos 3A and sin 3A, which are:

(cos 3A 6cos^3 A - 3cos A) (sin 3A 3sin A - 6sin^3 A)

Substituting these into the expression, we get:

(frac{3cos A (6cos^3 A - 3cos A)}{3sin A - (3sin A - 6sin^3 A)} frac{6cos^3 A}{6sin^3 A} cot^3 A)

Proving the Identities

To further validate the correctness of the formulas, let's prove that:

(cos 3A 4cos^3 A - 3cos A) (sin 3A 3sin A - 4sin^3 A)

Proving (cos 3A 4cos^3 A - 3cos A)

To prove: (cos 3A 4cos^3 A - 3cos A)

Let's start with the left-hand side (LHS) and use the compound angle formula:

(cos 3A cos 2Acos A - sin 2Asin A)

(cos 2A cos^2 A - sin^2 A)

(sin 2A 2sin Acos A)

Substitute these into the compound angle formula:

(cos 3A (cos^2 A - sin^2 A)cos A - (2sin Acos A)sin A)

(cos 3A cos^3 A - sin^2 Acos A - 2sin^2 Acos A)

(cos 3A cos^3 A - (1 - cos^2 A)cos A - 2(1 - cos^2 A)cos A)

(cos 3A cos^3 A - cos A cos^3 A - 2cos A 2cos^3 A)

(cos 3A 4cos^3 A - 3cos A)

The right-hand side (RHS) is also:

(4cos^3 A - 3cos A)

Therefore, LHS RHS, and the identity is proven.

Proving (sin 3A 3sin A - 4sin^3 A)

To prove: (sin 3A 3sin A - 4sin^3 A)

Let's start with the left-hand side (LHS) and use the compound angle formula:

(sin 3A sin 2Acos A cos 2Asin A)

(sin 2A 2sin Acos A)

(cos 2A cos^2 A - sin^2 A)

Substitute these into the compound angle formula:

(sin 3A (2sin Acos A)cos A (cos^2 A - sin^2 A)sin A)

(sin 3A 2sin Acos^2 A cos^2 Asin A - sin^3 A)

(sin 3A 3sin Acos^2 A - sin^3 A)

Using the identity (sin^2 A 1 - cos^2 A):

(sin 3A 3sin A(1 - sin^2 A) - sin^3 A)

(sin 3A 3sin A - 3sin^3 A - sin^3 A)

(sin 3A 3sin A - 4sin^3 A)