How do you find the exact value of Sin(13pi)?

since 13π = 12π + π, it suffices to get the exact value of sin(π).

adding multiples of 2π does not change the value of the sine function.

To find the exact value of sin(13π), you can use the properties of the sine function and the periodicity of trigonometric functions.

1. Start by finding the number of complete revolutions in 13π: Divide 13π by 2π (the period of sine function) to get the number of complete revolutions.

13π / 2π = 6.5

This means that sin(13π) has completed 6.5 revolutions around the unit circle.

2. Since sin function repeats itself after each complete revolution, we can ignore the full revolutions and focus on the remaining portion. In this case, we can ignore the 6 full revolutions, so we are left with 0.5 revolution (or π radians).

sin(13π) = sin(0.5π)

3. The sine function has certain values for specific angles. We can use the special angles to find the exact value of sin(0.5π).

sin(0.5π) = sin(π/2)

4. The value of sin(π/2) is equal to 1.

Therefore, the exact value of sin(13π) is 1.

To find the exact value of Sin(13π), we need to use the properties of the sine function and the unit circle.

The unit circle has a radius of 1 and is centered at the origin (0, 0) on the coordinate plane. It helps us understand the relationship between angles and trigonometric functions such as sine, cosine, and tangent.

In the case of the sine function, the value of sin(θ) represents the y-coordinate of the point on the unit circle for the given angle θ.

To find the exact value of Sin(13π), we need to convert the angle from radians to degrees. We know that 2π radians is equivalent to 360 degrees, meaning 1π radian is equal to 180/π degrees.

Therefore, 13π radians is equal to (13π) * (180/π) degrees.
Simplifying, we get 13 * 180 degrees.

Since the sine function is periodic with a period of 360 degrees, we can use this property to find the exact value of Sin(13π). We find that Sin(13π) is equivalent to Sin(13 * 180 degrees).

Next, we need to determine which quadrant the angle 13 * 180 degrees lies in. Since the angle is a multiple of 180 degrees, it will lie on the x-axis.

In the first and second quadrants, the y-coordinate is positive. Since the angle 13 * 180 degrees lies on the x-axis, the y-coordinate will be zero.

Therefore, the exact value of Sin(13π) is 0.