How do you evaluate the expression cos3pi

cos (3π)

= cos(2π + π) , ----> one rotation plus another π
= cosπ
= -1

To evaluate the expression cos(3π), we can use the unit circle or the periodicity property of the cosine function.

1. Using the Unit Circle:
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. To evaluate cosine at a specific angle, we draw a line from the origin to a point on the unit circle corresponding to that angle.

In this case, 3π is equal to 3 times the value of π (pi), which is approximately 3.14159. We divide this value by π to find the corresponding angle on the unit circle.

3π / π = 3

So, the angle we need to find on the unit circle is 3.

Now, we can look at the x-coordinate of the point corresponding to an angle of 3 on the unit circle, which gives us the value of cos(3π).

By examining the unit circle, we can see that the x-coordinate at an angle of 3 is -1.

Therefore, cos(3π) is equal to -1.

2. Using the periodicity property:
The cosine function has a periodicity of 2π, meaning that cos(x + 2π) = cos(x) for any value of x.

In this case, 3π can be expressed as 2π + π. Using the periodicity property, we can rewrite cos(3π) as cos(2π + π).

Since cos(2π) = 1 (cosine of a full circle) and cos(π) = -1 (cosine of half a circle), we can substitute these values into cos(2π + π) to simplify the expression.

cos(2π + π) = cos(2π) * cos(π) - sin(2π) * sin(π)
= 1 * (-1) - 0 * 0
= -1

Therefore, cos(3π) is equal to -1.