Yes, sin(nπ) is always zero for all integers n.
This is a fundamental property of the sine function. Let's explore why:
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The Sine Function and the Unit Circle: The sine function, sin(x), represents the y-coordinate of a point on the unit circle, where x is the angle measured counterclockwise from the positive x-axis.
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Integer Multiples of π: When n is an integer, nπ represents angles that are integer multiples of π radians. These angles correspond to points that lie on the x-axis of the unit circle.
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Points on the x-axis: Points on the x-axis always have a y-coordinate of 0. Since sin(x) is the y-coordinate on the unit circle, sin(nπ) = 0 for all integers n.
Examples:
- sin(0π) = sin(0) = 0
- sin(1π) = sin(π) = 0
- sin(2π) = sin(2π) = 0
- sin(-1π) = sin(-π) = 0
- sin(-2π) = sin(-2π) = 0
In summary, because integer multiples of π radians land on the x-axis of the unit circle, and the sine function represents the y-coordinate, sin(nπ) always equals zero when n is an integer.
