Sin(2x)

Definition

The expression sin(2x) represents the sine of an angle that is double the value of x. This function is essential in trigonometry, showing periodic behavior and transformations, which are crucial for solving various problems. Understanding sin(2x) allows for the application of trigonometric identities, particularly in simplifying expressions and differentiating functions using the chain rule.

5 Must Know Facts For Your Next Test

1. The derivative of sin(2x) can be calculated using the chain rule, resulting in 2cos(2x).
2. sin(2x) has a maximum value of 1 and a minimum value of -1, as is true for all sine functions.
3. The function sin(2x) completes one full cycle from 0 to π, meaning its period is π instead of the usual 2π for sin(x).
4. Graphically, sin(2x) oscillates faster than sin(x), which can be visualized by comparing their respective graphs.
5. Using the double angle formula, you can rewrite expressions involving sin(2x) for simplification in calculus problems.

Review Questions

• How does the chain rule apply to the differentiation of sin(2x), and what is the result?
  • When applying the chain rule to differentiate sin(2x), you treat the outer function as sin(u) where u = 2x. The derivative of sin(u) is cos(u), and then you multiply it by the derivative of u with respect to x, which is 2. Therefore, the derivative of sin(2x) is 2cos(2x). This highlights how the chain rule helps in dealing with composite functions.
• In what way does the double angle formula for sine provide insight into simplifying expressions involving sin(2x)?
  • The double angle formula states that sin(2x) = 2sin(x)cos(x). This relationship allows us to rewrite expressions involving sin(2x) into terms of sin(x) and cos(x), making it easier to work with especially in integration and solving trigonometric equations. It demonstrates how identities can simplify complex problems.
• Evaluate how understanding periodicity enhances the analysis of the function sin(2x) compared to sin(x).
  • Understanding periodicity is crucial when analyzing sin(2x), as it has a period of π, half that of sin(x), which has a period of 2π. This means that while both functions oscillate between -1 and 1, sin(2x) completes its cycles twice as fast. Recognizing this difference allows for better predictions about function behavior over specific intervals and aids in graphing them accurately.

Related terms

Double Angle Formula: A trigonometric identity that expresses sin(2x) in terms of sin(x) and cos(x), given by sin(2x) = 2sin(x)cos(x).

Chain Rule:

A fundamental differentiation technique that allows the differentiation of composite functions, such as when applying it to sin(2x) where you differentiate an outer function with an inner function.

Periodicity: The property of trigonometric functions like sin(x) and cos(x) to repeat their values in regular intervals, with sin(2x) having a period of π.