The double angle formulas for sine and cosine are powerful tools in trigonometry, allowing us to express \( \sin(2\theta) \) and \( \cos(2\theta) \) in terms of \( \sin(\theta) \) and \( \cos(\theta) \). These formulas are not just mathematical curiosities; they have practical applications in physics, engineering, and computer science. In this article, we will derive these formulas from first principles.
The Double Angle Formula for Sine
The double angle formula for sine is:
\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]
Derivation
To derive this formula, we start with the sum formula for sine:
\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]
Setting \( A = B = \theta \), we get:
\[
\sin(2\theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta)
\]
\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]
And thus, we have derived the double angle formula for sine.
The Double Angle Formulas for Cosine
The double angle formulas for cosine are:
- \( \cos(2\theta) = \cos^2(\theta) – \sin^2(\theta) \)
- \( \cos(2\theta) = 2\cos^2(\theta) – 1 \)
- \( \cos(2\theta) = 1 – 2\sin^2(\theta) \)
Derivation
We start with the sum formula for cosine:
\[
\cos(A + B) = \cos(A)\cos(B) – \sin(A)\sin(B)
\]
Setting \( A = B = \theta \), we get:
\[
\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
\]
This is our first formula. To derive the other two, we use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
- \( \cos^2(\theta) – \sin^2(\theta) = \cos^2(\theta) – (1 – \cos^2(\theta)) = 2\cos^2(\theta) – 1 \)
- \( \cos^2(\theta) – \sin^2(\theta) = (1 – \sin^2(\theta)) – \sin^2(\theta) = 1 – 2\sin^2(\theta) \)
Conclusion
We have successfully derived the double angle formulas for sine and cosine. These formulas are not just theoretical constructs but have real-world applications in various fields. Understanding them deeply enriches our understanding of trigonometry and, by extension, the universe itself.
