Introduction
The double angle formulas for sine and cosine are essential tools in trigonometry, with applications ranging from solving trigonometric equations to simplifying complex expressions. In this article, we will derive these formulas step-by-step to provide a deeper understanding of their origins.
The Angle Addition Formulas
Before diving into the double angle formulas, it’s crucial to understand the angle addition formulas for sine and cosine:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
\[
\cos(A + B) = \cos A \cos B – \sin A \sin B
\]
Deriving the Double Angle Formula for Cosine
Standard Form
Starting with the angle addition formula for cosine, \( \cos(A + B) = \cos A \cos B – \sin A \sin B \), we set \( A = B = \theta \):
\[
\cos(2\theta) = \cos \theta \cos \theta – \sin \theta \sin \theta
\]
\[
\cos(2\theta) = \cos^2 \theta – \sin^2 \theta
\]
Alternative Forms
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can express the formula in terms of either sine or cosine alone.
1. In terms of \( \cos \) alone:
\[
\cos^2 \theta = 1 – \sin^2 \theta
\]
\[
\cos(2\theta) = 1 – 2\sin^2 \theta
\]
2. In terms of \( \sin \) alone:
\[
\sin^2 \theta = 1 – \cos^2 \theta
\]
\[
\cos(2\theta) = 2\cos^2 \theta – 1
\]
Deriving the Double Angle Formula for Sine
Starting with the angle addition formula for sine, \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), we set \( A = B = \theta \):
\[
\sin(2\theta) = \sin \theta \cos \theta + \cos \theta \sin \theta
\]
\[
\sin(2\theta) = 2 \sin \theta \cos \theta
\]
Concluding Remarks
In the quest for understanding, the double angle formulas serve as humble reminders of the intricate tapestry of Creation. They are not merely tools for computation, but keys to unlocking the deeper harmonies of the universe. As St. Augustine might say, “Understanding is the reward of faith. Therefore, seek not to understand that you may believe, but believe that you may understand”1, and as Pascal would remind us, “The eternal silence of these infinite spaces frightens me.”2 In mastering these formulas, we not only advance in mathematical skill but also draw nearer to the Divine, confronting the awe-inspiring vastness and precision of the cosmos.
Sources
- St. Augustine, “Tractates on the Gospel of John,” Tractate 29, Section 6.
- Blaise Pascal, “Pensées,” Section 206.
