To find the values of sin2x, cos2x, and tan2x from the given information that cscx = 8 and tanx = 0, we start by interpreting these functions.
Cosecant is the reciprocal of sine. Thus, if cscx = 8, we have:
sinx = 1/cscx = 1/8
Now, knowing that sinx = 1/8 lets us find cosx using the Pythagorean identity:
sin²x + cos²x = 1
Substituting sinx:
(1/8)² + cos²x = 1
1/64 + cos²x = 1
cos²x = 1 – 1/64 = 63/64
Taking the square root, we get:
cosx = √(63/64) = √63/8
Next, since tanx = sinx/cosx, and we know tanx = 0, this means sinx must be 0. However, we earlier found that sinx = 1/8, which suggests that the given tanx value may not relate directly to the same angle x. Rather, let’s use our determined values of sinx and cosx:
Now, we can find sin2x and cos2x:
sin2x = 2sinx cosx
Substitute sinx and cosx:
sin2x = 2*(1/8)*(√63/8) = (√63/8)
cos2x = cos²x – sin²x
cos2x = (√63/8)² – (1/8)² = (63/64) – (1/64) = 62/64 = 31/32
Next, we find tan2x:
tan2x = sin2x/cos2x
tan2x = (√63/8) / (31/32) = (√63 * 32) / (8 * 31) = (4√63) / 31
In conclusion, we have:
sin2x = √63/8
cos2x = 31/32
tan2x = (4√63) / 31