The value of sin(π) is 0.
Understanding Sin(π) Using the Unit Circle
The unit circle provides a visual and intuitive way to understand trigonometric functions like sine. Here's how it applies to calculating sin(π):
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Unit Circle Basics: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane.
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Angles and Coordinates: An angle θ (theta) is measured counter-clockwise from the positive x-axis. The point where the terminal side of the angle intersects the unit circle has coordinates (cos θ, sin θ). Therefore, the y-coordinate of this point is sin θ.
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π Radians: π radians (or 180 degrees) represents a half-rotation around the unit circle. This means starting from the positive x-axis and rotating halfway around.
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Location at π: After rotating π radians, you arrive at the point (-1, 0) on the unit circle.
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Determining sin(π): Since the coordinates at π are (-1, 0), and the y-coordinate represents the sine of the angle, sin(π) = 0. The video excerpt confirms this: "We haven't really gone up or down so the y coordinate is zero." This refers to the fact that at π radians, the point on the unit circle has a y-coordinate of 0.
Summary
Using the unit circle, we can visually determine that sin(π) = 0 because at π radians, the corresponding point on the unit circle is (-1, 0), and the y-coordinate represents the sine value.
