if x is a positive acute angle, then express tan(180+x) as a function of x

recall your sum of angles formula for tan

recall that tan(180) = 0

To express tan(180+x) as a function of x, we can use the fact that the tangent function is periodic with a period of 180 degrees.

Since x is a positive acute angle, it lies in the first quadrant (0-90 degrees). Adding 180 degrees to x will result in an angle in the third quadrant (180-270 degrees), where the tangent function is negative.

We know that tangent is the ratio of sine to cosine. So, we can express tan(180+x) as:

tan(180+x) = -tan(x)

Therefore, tan(180+x) can be expressed as a function of x as -tan(x).

To express tan(180+x) as a function of x, we can use the trigonometric identity for tangent:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b))

In this case, a = 180 and b = x. Since x is a positive acute angle, it falls between 0 and 90 degrees.

First, let's find the values for tan(180) and tan(x).
Since tan(180) is undefined, we cannot directly calculate it. However, we know that tan(180) = sin(180) / cos(180).

To evaluate sin(180) and cos(180), we can use the unit circle:
- At 180 degrees, sin(180) = 0 and cos(180) = -1.

Therefore, tan(180) = sin(180) / cos(180) = 0 / (-1) = 0.

Now, let's find the value for tan(x).
Since x is a positive acute angle, we can use the properties of the unit circle to find sin(x) and cos(x).

Once we have the values for sin(x) and cos(x), we can compute tan(x):
tan(x) = sin(x) / cos(x).

Finally, we can substitute these values into the trigonometric identity:

tan(180 + x) = (tan(180) + tan(x)) / (1 - tan(180) * tan(x)) = (0 + tan(x)) / (1 - 0 * tan(x)) = tan(x) / 1 = tan(x).

So, tan(180 + x) = tan(x) in this case.