Sin(3θ)

Definition

sin(3θ) represents the sine of three times an angle θ and is a specific case of the sine function involving a multiple angle. This expression can be expanded using trigonometric identities to express it in terms of sine and cosine of θ, which makes it easier to analyze or solve equations involving multiple angles. Understanding sin(3θ) allows us to explore its properties and relationships with other trigonometric functions and enables the solving of equations that involve angles multiplied by integers.

5 Must Know Facts For Your Next Test

1. The formula for sin(3θ) can be derived from the angle addition formula, yielding sin(3θ) = 3sin(θ) - 4sin^3(θ).
2. When analyzing sin(3θ), one can also express it in terms of cosine using the identity sin^2(θ) + cos^2(θ) = 1.
3. The graph of sin(3θ) has a frequency three times that of sin(θ), resulting in three complete cycles within the same interval as one cycle of sin(θ).
4. The amplitude of sin(3θ) remains the same as that of sin(θ), which is 1, while the period decreases to 2π/3.
5. Solving equations that include sin(3θ) often involves utilizing its expanded form, which can simplify finding solutions across various quadrants.

Review Questions

• How can you derive the expression for sin(3θ) using trigonometric identities?
  • To derive sin(3θ), start with the angle addition formula. You can express sin(3θ) as sin(2θ + θ). Applying the formula gives us sin(2θ)cos(θ) + cos(2θ)sin(θ). Further expanding sin(2θ) and cos(2θ) using double angle identities results in a final expression: sin(3θ) = 3sin(θ) - 4sin^3(θ). This shows how sin(3θ) relates to simpler sine terms.
• Explain how the properties of sin(3θ) differ from those of sin(θ), particularly regarding amplitude and period.
  • While the amplitude of sin(3θ) remains at 1, like that of sin(θ), its period changes significantly. The period for sin(3θ) is 2π/3, meaning it completes three cycles in the same interval where sin(θ) completes one cycle (2π). This results in a more frequent oscillation within a given interval, showing how multiple angles impact the behavior of sine functions.
• Evaluate how understanding sin(3θ) aids in solving complex trigonometric equations and contributes to deeper mathematical concepts.
  • Understanding sin(3θ) helps break down complex trigonometric equations into more manageable parts. By expressing it through its derived form, such as 3sin(θ) - 4sin^3(θ), we can easily factor or isolate variables when solving. This knowledge extends to more complex mathematical concepts such as Fourier series and harmonic analysis, where multiple angles play a key role in representing periodic functions accurately.

Related terms

Trigonometric Identities: Equations that hold true for all values of the variable involved, which relate different trigonometric functions to one another.

Angle Addition Formula: Formulas used to find the sine, cosine, or tangent of the sum or difference of two angles.

Periodic Function: A function that repeats its values in regular intervals or periods, characteristic of all trigonometric functions.