Understanding the Odd Property of the Sine Function: A Derivation from the Unit Circle and Definition of Sine

sin(-x) =

The sine function is an odd function, which means that it satisfies the property sin(-x) = -sin(x) for any angle x

The sine function is an odd function, which means that it satisfies the property sin(-x) = -sin(x) for any angle x. This property can be derived from the unit circle or using the definition of the sine function.

Using the unit circle:
The unit circle is a circle with a radius of 1 centered at the origin (0,0) in a coordinate plane. To find the sine of an angle, we draw a line from the origin to the point on the unit circle that corresponds to the angle (in radians). The y-coordinate of this point gives us the value of sine for that angle.

Since the unit circle is symmetric about the x-axis, any angle and its opposite have the same y-coordinate. This means that sin(x) = sin(-x). However, the sign of the y-coordinate will be different for the two angles. Therefore, sin(-x) = -sin(x).

Using the definition of the sine function:
The sine function can also be defined using the right triangle. Suppose we have a right triangle with an angle x. Let’s label the opposite side to angle x as “o” and the hypotenuse as “h”. Then, sin(x) = o/h.

Now, if we consider the angle -x, the opposite side will still be “o” because the orientation of the triangle does not change. However, the original hypotenuse will be the adjacent side to -x, which we’ll label as “a”.

Since the hypotenuse of a right triangle is always longer than the opposite side, we have h > o. Therefore, a > o and sin(-x) = o/a < o/h = sin(x). But since the sine function is negative in the fourth quadrant of the unit circle, we have sin(-x) = -sin(x). In conclusion, sin(-x) = -sin(x) for any angle x.

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