$A$ and $B$ are both obtuse angles such that $\sin(A) = \frac {5} {13}$ and $\tan B = \frac {-3} {4}$. Find exact values for $\sin (A + B)$.

Assuming that the pythagorean theorem is used to answer the question, how is it possible that the values for $\tan B = \frac {-3} {4}$? (That is, a $3$–$4$–$5$ right angle triangle… how can a side be negative?)

## Best Answer

That is a great question that is better answered if you understand that all the trigonometric functions have different ways of being defined. The natural, I'd say, historic, way, is the one to which you refer. A sine of an angle is the ratio of its opposite side to the hypotenuse, its cossine is the ratio of its adjacent to the hypotenuse, etc. In that definition, intimately tied to triangles and Euclidian geometry, trigonometric functions are alway positive as they are defined using real positive lengths. But imagine your question had nothing to do with triangles, or even shapes, and you were dealt with trigonometric functions of which solution you knew couldn't be a positive number? Would you just throw away these handy functions just because they are restricted to yielding positive numbers? Of course not. So you just define them so that they can output negative numbers as well. Namely, if you have a point of which coordinates are real numbers (x,y), you define the sine as $\frac{y}{\sqrt{x^2+y^2}}$. This is still inspired by the triangle definition, but now, as y can be a negative number, so can sine. The same for the other trig functions.

So, to answer your question concretely. How can a side be negative? It can't, but that is besides the question. Your problem is not to find such a triangle, it is to find the value of a function ( $\sin{(A+B)}$). That function can be negative, and, as such, so can the tangent.

As you go deeper and deeper into mathematics you will find that the way you used to think about some elementary concept morphs and evolves and grows out of its initial meaning. Just like most likely the first way you defined squaring something was as the area with length "something", but now you use squaring without ever needing to refer to squares. If I ask you to solve, for example, $x^2+1=0$ you will find that there is no square of length $x$ that satisfies this condition. However there is a sense in which this function has a solution, and the re-definition of squaring something eventually leads to complex numbers, which you might have heard of. Trig functions are such elementary concepts, and some representations would even leave you wondering if that really is a sine, or a cossine. To leave you with one that might shock you just enough for you to get interested but not so much that you instantly give up trying to understand it, I present you this: $$\sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+...$$ I know that you only asked about angles and lengths, but I feel you can take away some meaningful intuition from all this :)