# Thread: Drawback of Sine Rule

1. ## Drawback of Sine Rule

I found that applying sine rule has a disadvantage, especially when you want to find one of the angles and the angle is an obtuse angle.

For example, consider a triangle ABC
Angle B = 30 degrees

AB = AC = 1 cm
$\displaystyle BC=\sqrt{3}$ cm

The problem arises here,
I tried to use sine rule to get Angle A,
$\displaystyle \frac{BC}{\sin{A}}=\frac{AC}{\sin{B}}$
$\displaystyle \sin{A}=\frac{\sqrt{3}}{2}$
$\displaystyle Angle A=60^{o}$

But in fact, angle A should be 120 degrees, because the calculator will give an acute angle instead of obtuse angle.

Is there any way to avoid this kind of mistake

2. Avoid what mistake?
The Law of Sines is always correct.

In your example, the angle A cannot be 60 degrees.
[Since the triangle is isosceles, with the base angles at 30 degrees each, then the 3rd angle, angle A, cannot be 60 degrees only.

sinA = sqrt(3) / 2.
Then A is not 60 degrees only.
Sine is positive in the 2nd quadrant too, so A = 120 degrees there.
Given the choice A=60deg or A=120deg, which one fits your triangle?

3. Draw a diagram. If it doesn't look right then you're either a bad artist or you have the wrong angle.

4. Originally Posted by DivideBy0
If it doesn't look right then you're either a bad artist or you have the wrong angle.
haha. funny you should say that. i tutor math at school and in my experience, most math students terrible artists and are horrible at drawing diagrams.

5. I didn't say that Sine Rule is wrong.

The triangle I used above is just an example, I should have a triangle which is not isosceles.

My question is that if angle A is obtuse, then we will have to be careful if we find angle A by Sine rule, because the calculator will give the angle in the first quadrant. However, Cosine rule will not. Isn't that a disadvantage of using Sine Rule??

I understand that we may draw diagram to see whether the angle is acute or obtuse, but what if Angle A is an angle between 90 and 100 degrees, that will be difficult to visualise, right? Unless you have super eyesight.

6. Originally Posted by acc100jt
I didn't say that Sine Rule is wrong.

The triangle I used above is just an example, I should have a triangle which is not isosceles.

My question is that if angle A is obtuse, then we will have to be careful if we find angle A by Sine rule, because the calculator will give the angle in the first quadrant. However, Cosine rule will not. Isn't that a disadvantage of using Sine Rule??

I understand that we may draw diagram to see whether the angle is acute or obtuse, but what if Angle A is an angle between 90 and 100 degrees, that will be difficult to visualise, right? Unless you have super eyesight.
Reason why the Law of Cosines will always give you the correct angle is because cosine is positive in the 1st quadrant and is negative in the second quadrant. Meaning, in using the Law of Cosines, any positive cosine value will give you an acute angle always, while any negative cosine value will give you an obtuse always.

In using the Law of sines, the sine is positive both in the 1st and 2nd quadrants. That's why any sine value would either be for an acute or obtuse angle.
If you cannot discern the angle via your sketch, then solve for the other, 3rd, angle too. Maybe by then you'd know which one is acute or obtuse.
If that fails too, check with the Law of Cosines. As you said, that will always give you the correct angle.

You never said that the Law of Sines is wrong, yes, but you mentioned about a mistake in using it. There can never be a mistake.

7. C'mon, we don't need trig. here, just drop an altitude.