Final answer:
To evaluate tan 2x given that cos x = √2/2 in the fourth quadrant, we use the double-angle formula for tangent and the Pythagorean identity to find that tan 2x is undefined since the calculation involves a division by zero.
Step-by-step explanation:
If cos x = √2/2 and x is a fourth quadrant angle, we need to determine the value of tan 2x. In the fourth quadrant, cosine is positive, but sine is negative. This is important because the tan function is the quotient of sin over cos.
Using the double-angle formula for tangent, we get:
- tan 2x = 2 tan x / (1 - tan2 x)
Since we know cos x, we can find sin x from the Pythagorean identity sin2 x + cos2 x = 1. This gives us sin x = -√(1 - cos2 x) because sin is negative in the fourth quadrant. Substituting the given cos x value, we find sin x to be -√(1 - (√2/2)2).
Now we can calculate tan x as sin x / cos x = -√(1 - (√2/2)2) / (√2/2). This simplifies to just -1. Putting this into our double-angle formula yields:
- tan 2x = 2(-1) / (1 - (-1)2)
- tan 2x = -2 / 0
Here we see the denominator becomes 0, which makes tan 2x undefined.