Proving the Double Angle Identity for Sine: sin 2x 2 sin x cos x
The double angle identity for sine, sin 2x 2 sin x cos x, is a fundamental concept in trigonometry. It has a number of practical applications in calculus and is used extensively in various fields, including physics and engineering.
Proof Using Angle Addition Formula
A common method to prove this identity is by using the angle addition formula for sine. According to the sine addition formula:
(sin(a b) sin a cos b cos a sin b)
Let's substitute (a x) and (b x). Thus:
[sin(2x) sin(x x) sin x cos x cos x sin x]This simplifies to:
[sin(2x) 2 sin x cos x]Geometric Proof Using Unit Circle
We can also prove the identity using a geometric approach involving a unit circle. The unit circle is a circle with a radius of 1, centered at the origin. Let’s construct this geometrically:
Step 1: Draw a diameter AOA on the unit circle. Take a point P on the circle and join OP, PA, and PA’. Let ∠POA be (2θ).
Step 2: Drop a perpendicular from P onto AA’ and let N be the foot of the perpendicular. Note that ∠APA is a right angle, and thus triangle A’PA can be broken into two right-angled triangles: APN and PNA.
Step 3: In the right-angled triangle OPN, based on the definition of sine:
[PN/OP sin 2θ]Given that (OP 1) unit (since it’s a unit circle), we have:
[PN sin 2θ]Step 4: From triangle A’PA, we use the definition of cosine:
[PA’ A’A cos θ]Since (A’A 2) units (it’s the diameter), we get:
[PA’ 2 cos θ]Step 5: From triangle APN, we again use the definition of sine:
[PN PA’ sin θ]Step 6: Combining equations 2 and 3, we get:
[PN 2 cos θ sin θ]Step 7: Equating equations 1 and 4, we find:
[sin 2θ 2 cos θ sin θ]Complex Numbers Proof
We can also prove the double angle identity for sine using complex numbers. Starting with the complex exponential form of sine and cosine:
[sin x frac{e^{ix} - e^{-ix}}{2i}] [cos x frac{e^{ix} e^{-ix}}{2}]Substitute these into the expression for (2 sin x cos x):
[begin{align*} 2 sin x cos x 2 left(frac{e^{ix} - e^{-ix}}{2i}right) left(frac{e^{ix} e^{-ix}}{2}right) frac{1}{2i} left(e^{2ix} - e^{-2ix}right) sin 2x end{align*}]Further Insights Using De Moivre’s Theorem
De Moivre's Theorem for (n2) states: [(cos x i sin x)^2 cos 2x i sin 2x]
Expanding the left-hand side:
[(cos x i sin x)^2 cos^2 x - sin^2 x 2 cos x sin x i]Equate the imaginary parts of both sides:
[sin 2x 2 sin x cos x]This confirms the double angle identity for sine once again.
Understanding these proofs not only solidifies the knowledge of the double angle identity but also enhances the understanding of trigonometric identities in general. It showcases the interconnectedness of various branches of mathematics, including geometry, algebra, and complex numbers.