Question

Is the function tan t positive or negative in Quadrant IV?

Answer 1

Answer:

negative

Step-by-step explanation:

hey there,

< First, let's find out what Quadrant IV is. IV means 4. The 4th quadrant is the one at the lower right corner.

To find tangent, you have to draw a triangle in the quadrant. If you draw a triangle (like pictured below), you can find out each line. The way you find this is by marking the adjacent, opposite, and hypotenuse. I hope you already understand what each of those are, but if you don't, feel free to ask me. As we can tell here, the angle is the one touching the intersection of the graph.

Tan = O/A ($\frac{opposite}{adjacent}$)

The opposite line of the triangle would be marked as "x" in this triangle, and adjacent which is the line right next to the angle (but isn't the hypotenuse), is here marked as "$\sqrt{1-x^2}$".

As you can see here, in each quadrant, the triangle is a little different in terms of positive/negative. Remember: *all the lines that are going right or up are always positive, and the ones going down or left are negative.*

Here, we can see that the opposite line is going down, meaning it's NEGATIVE. The adjacent line is going right, though, so it's POSITIVE. Now let's put those into the equation.

Tan = O/A = negative/positive

Pretend that O is equal to 2 and A is equal to 1. Here, we can see O is negative so it would be -2 and A is positive so it remains 1. Divide the two. -2/1 = -1. You can see the result is NEGATIVE. That means your answer would be negative! (Use the simple numbers method if you forget!) So as you can tell, if it was positive/positive, the answer would be positive and etc. >

Hope this helped! Feel free to ask anything else.

Answer 2

If you draw the circle and divide into 4 quadrants, you can see that the sine is negative, since the quadrant is situated in the negative part of YY', and that the cosine is positive, since the quadrant is situated in the positive part of XX'.
And $tan = \frac{sin}{cos}$ , and negative divided by positive gives a negative answer, which means that the tangent is negative on the quadrant IV.
Hope this helps :)