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.