Math CentralQuandaries & Queries


Question from jonathan, a student:

how can i solve this if the only given is sin theta= -3
tan theta=
cos theta=
cot theta=
co secant theta=
secant theta=
can you teach me and show how to solve??

Hi Jonathan. That question has no answer. Look at a sine wave: all the values of f(x) = sin(x) vary between -1 and 1, because that's the range of the sine function. Your question says sin(theta) = -3, so that's not possible.

But, if you meant to say sin(theta) = -0.3, this is how you would solve it.

The sine is the ratio of the opposite side to the hypotenuse. Consider a "unit circle" (that's a circle with radius 1). The radius is the hypotenuse, so in this case, the opposite side is -0.3. That's the "y" value on a normal graph, which means either quadrant III or quadrant IV. There are in fact two angles (two values of theta) in which sin(theta) = -0.3.

Now if the hypotenuse of a right triangle is 1 and the opposite side is 0.3 (we can ignore the negative sign for the moment), then Pythagoras can tell us the length of the adjacent (x) side. 0.32 + x2 = 12. When I use my calculator to solve this, I get x = ± 0.95.

Now we know:
y = opposite = -0.3 and
x = adjacent = +/- 0.95 and
r = hypotenuse = 1

We can calculate any of the trigonometric ratios, just be considering their definitions:
tan = opposite / adjacent.
cos = adjacent / hypotenuse.
cot = adjacent / opposite.
csc = hypotenuse / opposite.
sec = hypotenuse / adjacent.

So you can find all the values. Just remember that some of these will have two answers (because of the +/- on the adjacent length) and some will only have one value (those that don't use the adjacent side).

Hope this helps,
Stephen La Rocque.

About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS