**A quick note before we start, remember that the unit circle is 360 degrees. When something asks for the "least possible positive measure", it's just asking for something between 0 and 360 degrees.
Let's break it down into parts.
I find it's best to visualize what we're actually doing. Therefore, we're first going to draw the right triangle the question gives you.
- Plot the point (5, -12) on a graph.
- Draw a line from the origin, also known as point (0, 0) on your graph, to the point you just plotted at (5, -12). This line you just drew gives you the hypotenuse side of the triangle you're about to draw.
- Now draw a line from the origin (0, 0) to point (5, 0)
- Now draw another line from point (5, 0) to the original point you plotted (5, -12).
At the moment you should have a nice right triangle drawn. You may have noticed that you also have the side lengths of your triangle, given by the x and y values. The x-value is 5 and the y value is -12. With that we can now find the length of the hypotenuse.
Using the Pythagorean Theorem we know that: a-squared + b-squared = c-squared. So now we just plug in the numbers.
5(5) + -12(-12) = 25 + 144 = 169
c-squared = 169
We now take the square root of 169 to come up with 13. We now have the length of the hypotenuse at 13. Now we have all of the values we need to determine our trigonometric functions. I don't know if you're familiar with this:
Soh Cah Toa
Cho Sha Cao
It basically means that:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Cosecant = Hypotenuse / Adjacent
Secant = Hypostenuse / Adjacent
Cotangent = Adjacent / Opposite
Recall that the opposite value is the same thing as the y-value, which in this problem is -12, and the adjacent value is the x-value, which in this problem is 5, and we determined the hypotenuse value to be 13 or:
Opposite = -12
Adjacent = 5
Hypotenuse = 13
Now that we've defined the six trigonometric functions, and have our values we just plug in the numbers:
Sine = -12/13
Cosine = 5/13
Tangent = -12/5
Cosecant = 13/-12 or -13/12
Secant = 13/5
Cotangent = 5/-12 or -5/12
Remember that a ratio is just a fraction. So when it's asking if the ratio is positive or negative, that's just another way of asking "is the fraction positive or negative?".
So if point (x,y) is in quadrant 2, we know that the x-value will be positive and the x value will be negative.
Going back to our Soh Cah Toa Cho Sha Cao we can just plug in values. Remember that the y value is the Opposite and the x-value is the Adjacent.
Sine = y/h making this a positive ratio
Cosine = -x/h making this a negative ratio
Tangent = y/-x or -y/x making this a negative ratio
Cosecant = h/y making this a positive ratio
Secant = h/-x or -h/x making this a negative ratio
Cotangent = -x/y making this a negative ratio