Proving the Trigonometric Identity: cosA - √3 sinA 2 cos(Aπ/3)

Introduction

In the realm of trigonometry, proving identities involves manipulating expressions to show that they are equivalent. One common task is to demonstrate that certain trigonometric expressions can be simplified to a known, simpler form. This article will walk you through a detailed step-by-step proof of the identity: cosA - √3 sinA 2 cos(Aπ/3).

For reference, I have extensive teaching experience of more than 7 years, making it possible to explain this concept clearly and thoroughly.

Step-by-Step Proof

Let's start with the left side of the identity: cosA - √3 sinA.

Step 1: Factoring by Multiplying by 2/2

First, we multiply both terms by 2/2 to make the coefficients easier to handle:

[ text{cosA - √3 sinA} left(2frac{1}{2}right)text{cosA} - left(2frac{1}{2}right)sqrt{3}text{sinA} ]

Simplifying this, we get:

[ 2frac{1}{2}text{cosA} - 2frac{1}{2}sqrt{3}text{sinA} ]

Step 2: Substituting Known Trigonometric Values

We recognize that 1/2 corresponds to cos60° and √3/2 corresponds to sin60°. Thus, we can substitute these values:

[ 2frac{1}{2}text{cosA} - 2frac{1}{2}sqrt{3}text{sinA} 2text{cos60°cosA} - 2text{sin60°sinA} ]

Step 3: Applying the Cosine Addition Formula

Recall the cosine addition formula: cos(A B) cosAcosB - sinAsinB. We aim to match the form of this identity:

[ 2(text{cos60°cosA} - text{sin60°sinA}) ]

We can now compare the two expressions:

Given: 2(cos60°cosA - sin60°sinA)

And: cos(AB) cosAcosB - sinAsinB

Matching the forms, we get:

[ text{A} A quad text{and} quad text{B} 60° ]

Hence, substituting B 60° in the cosine addition formula, we have:

[ 2text{cos60°cosA - sin60°sinA} 2text{cos(60° A)} ]

Since 60° π/3 radians, we can rewrite this as:

[ 2text{cos(A π/3)} ]

Conclusion

Therefore, we have proven that

[ text{cosA - √3sinA} 2text{cos(Aπ/3)} ]

This identity is a powerful tool in simplifying complex trigonometric expressions, making the solution of equations and the evaluation of trigonometric functions much more straightforward.

With over 7 years of teaching experience, I can confidently say that understanding these core concepts is an essential step in mastering advanced topics in trigonometry.