To determine the values of cos(x) and tan(x) when sin(x) is given as 12, we first need to clarify an important aspect: the sine function, sin(x), ranges from -1 to 1 for all real values of x. This means that sin(x) cannot equal 12 because it exceeds the maximum possible value. Therefore, in standard mathematics, this situation leads to an inconsistency.
However, if we were to entertain a hypothetical scenario where we consider a value for sinusoidal behavior beyond the normal range, we could discuss the mathematical definitions of cos(x) and tan(x) in terms of sine.
Generally, from the Pythagorean identity, we know:
sin2(x) + cos2(x) = 1.
Since sin(x) is proposed as 12, we can plug this value into the identity:
122 + cos2(x) = 1.
Calculating that, we find:
144 + cos2(x) = 1.
This leads us to:
cos2(x) = 1 – 144.
So:
cos2(x) = -143.
Since the square of a cosine function cannot be negative, we conclude that such a sine value is impossible under real number parameters.
If we were considering complex numbers or alternative mathematical frameworks, further analysis could be performed, but within the realm of real numbers, such an approach yields no valid cosine or tangent values.
To summarize, if sin(x) is 12, it is mathematically incorrect, and thus we cannot derive meaningful values for cos(x) or tan(x) from this assumption.