Evaluate exactly cos(-(45pi/4))

11.25 * pi

that is 2 pi*5 + 1.25 pi
or five circles plus pi plus pi/4 CLOCKWISE
quadrant 2
cos is negative in quadrant two
so negative (1/2)sqrt 2

To evaluate cos(-(45π/4)), we can use the properties of the cosine function and the unit circle.

First, let's find an equivalent angle in the range [0, 2π) by adding 2π to the angle -(45π/4).

angle = -(45π/4) + 2π

Simplifying the expression:

angle = -(45π/4) + 8π/4 = (7π/4)

Now, cos(7π/4) can be found by considering the unit circle.

In the unit circle, the angle 7π/4 is in the third quadrant.

In the third quadrant, the x-coordinate of a point on the unit circle represents the cosine of the angle.

In the third quadrant, the angle 7π/4 corresponds to the point (-√2/2, -√2/2) on the unit circle.

Therefore, cos(7π/4) = -√2/2.

Thus, cos(-(45π/4)) is equivalent to -√2/2.

To evaluate cos(-(45pi/4)), we can use the properties of the cosine function and reference angles.

First, let's find the equivalent angle within the first revolution by adding or subtracting a multiple of 2pi.

Since -(45pi/4) is a negative angle, we can add 2pi to get an equivalent angle:
-(45pi/4) + 2pi = -(45pi/4) + (8pi/4) = (7pi/4)

Now, (7pi/4) is within the first revolution, so we can evaluate cos(7pi/4) directly.

To find the value of cos(7pi/4), we need to determine the reference angle. The reference angle in the fourth quadrant is obtained by subtracting the given angle from 2pi.

Reference angle = 2pi - (7pi/4) = (8pi/4) - (7pi/4) = (pi/4)

The cosine function is positive in the fourth quadrant. Therefore, the value of cos(7pi/4) is the same as the value of cos(pi/4), which is a commonly known angle.

The value of cos(pi/4) is √2/2.

Hence, the exact value of cos(-(45pi/4)) is √2/2.