The trigonometric identity for sin2x is a fundamental concept in mathematics. According to the provided reference, sin 2x is equal to twice the product of the sine and cosine functions of x.
Therefore, sin 2x = 2 sinx cosx.
Understanding the Formula
The formula, sin 2x = 2 sinx cosx, is a double-angle identity derived from trigonometric sum-to-product identities. It allows for the calculation of the sine of twice an angle using the sine and cosine of the single angle.
Key Elements
- sin 2x: This represents the sine of an angle that is twice the size of angle x.
- sinx: This is the sine of angle x.
- cosx: This is the cosine of angle x.
- 2: The result is that the product of sinx and cosx is multiplied by 2.
Table of Trigonometric Identities
| Identity | Formula |
|---|---|
| sin 2x | 2 sinx cosx |
| cos 2x | cos²x - sin²x or 2cos²x - 1 or 1 - 2sin²x |
| tan 2x | 2 tanx / (1- tan²x) |
Practical Implications
This identity has a wide range of applications:
- Calculus: Simplifying trigonometric integrals and derivatives.
- Physics: Analyzing wave phenomena and oscillatory motion.
- Engineering: Modeling signal processing and mechanical vibrations.
- Geometry: Solving complex geometric problems.
Example
Let's say we want to find the value of sin 2x when x is 30 degrees (π/6 radians).
- We know that sin(30°) = 1/2
- We know that cos(30°) = √3/2
- Substituting these values into the formula, we get sin(230°) = 2 (1/2) (√3/2)*
- sin(60°) = √3/2
Therefore, the identity sin 2x = 2 sinx cosx holds true.
Summary
The identity sin 2x = 2 sinx cosx is crucial for trigonometric calculations and problem-solving in various scientific disciplines. It is based on fundamental relationships within trigonometry and the trigonometric sum-to-product identities.
