Use the even odd properties to find the exact value of the expression. It may be helpful to draw the unit circle graph sin(-pi)
The even-odd property for sine states that sin(-x) = -sin(x).
sin(-pi) = -sin(pi)
The point on the unit circle at pi (or 180 degrees) is (-1, 0). Sine corresponds to the y-value, so sine at that point is 0.
-sin(pi) = -0 = 0
To find the exact value of the expression sin(-π) using the even-odd properties, let's start by considering the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin (0,0) in a coordinate plane. It is commonly used to understand the properties and values of trigonometric functions.
To find sin(-π), imagine drawing a line from the origin to a point on the unit circle that corresponds to an angle of -π (an angle formed by extending an imaginary line from the positive x-axis to the point on the unit circle). In this case, -π represents an angle that lies on the negative x-axis.
Now, let's consider the even-odd properties of the trigonometric functions:
1. Even property: If an angle θ is replaced by its negative counterpart (-θ), the values of cosine (cos) and secant (sec) remain the same.
2. Odd property: If an angle θ is replaced by its negative counterpart (-θ), the values of sine (sin), tangent (tan), cosecant (csc), and cotangent (cot) change sign.
Since sin is an odd function, we can use the odd property to determine the value of sin(-π). The value of sine will be negative if the angle is in the third or fourth quadrants of the unit circle.
In this case, sin(-π) corresponds to the angle -π, which lies on the negative x-axis in the third quadrant. By the odd property, sin(-π) will have the same magnitude as sin(π), but with a negative sign.
Referring to the unit circle, we know that sin(π) = 0, since π is the angle formed by the positive x-axis and the point (-1, 0) on the unit circle. Therefore, sin(-π) will have the same magnitude of 0 but with a negative sign.
So, the exact value of sin(-π) is -0 or simply 0.