To find the values of sin(8) and tan(8), we can start by analyzing the given information. We know that cos(8) = √2/2, which corresponds to specific angles in the unit circle.
Since √2/2 is the cosine of both π/4 (45 degrees) and 7π/4 (315 degrees), we need to determine which angle 8 is near, given the constraints 3π/2 < 8 < 2π. This means 8 is in the fourth quadrant of the unit circle, where the cosine is positive and both sine and tangent take particular signs.
In the fourth quadrant, the sine value is negative, and the tangent is also negative since sine and cosine in this quadrant have opposite signs.
For cosine, if cos(8) = √2/2, we can denote this angle (in the fourth quadrant) as:
8 = 7π/4
Next, we find sin(8) using the sine value corresponding to 7π/4:
sin(7π/4) = -√2/2.
Now, using the values of sine and cosine, we calculate tan(8):
tan(8) = sin(8) / cos(8) = (-√2/2) / (√2/2) = -1.
In summary, the values are:
- sin(8) = -√2/2
- tan(8) = -1