Question

We study about trig.ratios of sum and diff. Of two angles but an obtuse angle can never have right triangle so how can we fix the ratios?

Answer 1

The trigonometric ratios are signed, so work in all four quadrants. This is one of the ways trig makes something more complicated; we're constantly worried about quadrants and signs and whether to write `pm`.

The good part about the signs is that we can handle obtuse angles and negative angles with no special effort. If we stick to triangle angles for a moment, between 0 and 180 degrees, we notice that the sine is always positive, so doesn't distinguish between an acute angle and its obtuse supplement. The value of the cosine of a triangle angle, on the other hand, uniquely determines the angle. It's yet another reason for the rule of thumb when there's a choice, choose cosine.

Let's look at the cosine sum formula:

`cos(A+B)=cos A cos B - sin A sin B`

There's a minus sign; this quantity can be positive or negative or zero even if A and B are in the first quadrant. The negative case indicates the sum is an obtuse angle (or its negation).

For the angles not in the first quadrant, we allow negative sides yielding negative trig ratios. A negative cosine is an obtuse angle (second or third quadrant); a negative sine is a negative angle, third or fourth quadrant.