The tangent function is considered an odd function. This means that it satisfies the condition that tan(-x) = -tan(x) for all values of x in its domain.
To understand this, let’s recall the definition of an odd function: a function f(x) is odd if for every x in the function’s domain, the equation f(-x) = -f(x) holds true. For the tangent function, if we take an angle -x, we find that:
1. The tangent of -x is equal to the sine of -x divided by the cosine of -x.
2. Since sine is an odd function (sin(-x) = -sin(x)), and cosine is an even function (cos(-x) = cos(x)), we can expand it as:
tan(-x) = sin(-x) / cos(-x) = -sin(x) / cos(x) = -tan(x)
This confirms that the tangent function is indeed odd, as we have demonstrated that tan(-x) = -tan(x).
In conclusion, since tangent satisfies the criteria of an odd function, we can confidently state that tangent is an odd function.