The measure 0 of an angle in standard position is given. Find the exact values of cos 0 and sin 0 for the angle measure. Show your all your work.
7π/4
To find the exact values of cos (7π/4) and sin (7π/4), let's first determine the reference angle.
The reference angle is the acute angle between the terminal side of the angle and the x-axis.
To find the reference angle, subtract π from the given angle:
Reference angle = 7π/4 - π = 3π/4
Since the terminal side of the angle is in the fourth quadrant, both cos (7π/4) and sin (7π/4) will be negative.
Now, let's find the exact values of cos (3π/4) and sin (3π/4).
The unit circle can help us determine these values:
First, let's determine the coordinates of the point that corresponds to the reference angle on the unit circle.
The reference angle of 3π/4 corresponds to the point (-√2/2, √2/2) on the unit circle.
Therefore, cos (3π/4) = x-coordinate = -√2/2
and sin (3π/4) = y-coordinate = √2/2
Since both cos (7π/4) and sin (7π/4) will be negative:
cos (7π/4) = -(-√2/2) = √2/2
sin (7π/4) = - (√2/2) = -√2/2
Therefore, the exact values of cos (7π/4) and sin (7π/4) are √2/2 and -√2/2, respectively.