Trigonometric ratios can be used to find the sides of a right triangle given an angle and a side length. But what if we don't have a right triangle? What if the angle we're given is obtuse, which means it's greater than 90 degrees?
Well, we can still use trigonometric ratios to find the sides of an obtuse triangle, but we have to use some special rules. Let's take a look at an example:
Suppose we have an obtuse angle θ, and we know that the side opposite θ has length a. We want to find the length of the adjacent side (which we'll call b) and the length of the hypotenuse (which we'll call c).
First, we'll draw a line perpendicular to the side of length a, so that it creates a right triangle with angle θ. Then we can use the same trigonometric ratios that we use for acute angles, but we have to be careful with the signs.
The sine of an obtuse angle is still defined as the ratio of the opposite side to the hypotenuse, but since the angle is obtuse, the opposite side and the hypotenuse will have opposite signs. So we have sin(θ) = -a/c.
The cosine of an obtuse angle is defined as the ratio of the adjacent side to the hypotenuse, but again, we have to be careful with the signs. Since the angle is obtuse, the adjacent side and the hypotenuse will have the same sign. So we have cos(θ) = b/c.
Finally, the tangent of an obtuse angle is defined as the ratio of the opposite side to the adjacent side, but this time both sides will have negative signs. So we have tan(θ) = -a/b.
Using these ratios, we can solve for the unknown side lengths b and c. For example, we can rearrange the cosine ratio to get c = b/cos(θ), and then substitute the sine ratio to get -a/cos(θ) = -a/(-a/sin(θ)) = sin(θ), so c = a/sin(θ).
So even though the trigonometric ratios are defined in terms of acute angles, we can still use them to find the sides of an obtuse triangle by being careful with the signs.