Understanding the Cosine of Double Angle: cos2x, cosAcosB - sinAsinB Formula, and Equivalent Forms

Introduction

In trigonometry, the cosine of double angle, denoted as cos2x, is a fundamental concept used in various mathematical and engineering applications. This article explores the derivation and significance of cos2x using the cosine addition formula, cosAcosB - sinAsinB, and provides several equivalent forms of this expression.

Cosine Addition Formula

The cosine addition formula states that:

cos(A B) cosA cosB - sinA sinB

This formula is crucial for understanding the trigonometric relationships between angles. To find the cosine of double angle cos2x, we express 2x as the sum of two identical angles, i.e., A B 2x, where A x and B x.

Deriving cos2x

Substituting A x and B x into the cosine addition formula, we have:

cos(2x) cos(x x) cos(x)cos(x) - sin(x)sin(x)

This simplifies to:

cos(2x) cos^2(x) - sin^2(x)

This is one of the standard forms for cos2x and will be used extensively in trigonometric calculations.

Alternate Forms of cos2x

Using the Pythagorean identity, sin^2(x) cos^2(x) 1, we can derive additional forms of cos2x as follows:

Form 1: Using sin^2(x)

Rewrite the identity by substituting sin^2(x) 1 - cos^2(x):

cos(2x) cos^2(x) - (1 - cos^2(x))

Simplifying this expression:

cos(2x) cos^2(x) - 1 cos^2(x) 2cos^2(x) - 1

Form 2: Using cos^2(x)

Similarly, we can use the identity cos^2(x) 1 - sin^2(x):

cos(2x) (1 - sin^2(x)) - sin^2(x)

Simplifying this:

cos(2x) 1 - sin^2(x) - sin^2(x) 1 - 2sin^2(x)

Thus, we have three equivalent forms of cos2x:

cos2x cos^2(x) - sin^2(x) 2cos^2(x) - 1 1 - 2sin^2(x)

Conclusion

The cosine of double angle, cos2x, is a versatile and powerful trigonometric concept. Its derivations using the cosine addition formula and the Pythagorean identity provide a solid foundation for further mathematical exploration. Understanding these forms is crucial for solving complex trigonometric problems and is widely applicable in various scientific and engineering fields.

References:

1. Math Is Fun - Cosine of Double Angle

2. Khan Academy - Proof of Double Angle Identities