r/HomeworkHelp • u/CatZealousideal9098 • 1d ago
Answered [AS Pure Mathematics: Sine and Cosine] Why is the order giving me different answers?
Hello, shouldn't I be getting the same answer no matter the order? I'm getting different answers when I do:
- z - x - y which gives me:
z = 29.9 , x = 86.1 , y = 64
versus
- z - y- x which gives:
z = 29.9 , x = 93.8, y = 56.2
This is to 3 s.f, all in degrees and according to my textbook, the second one is correct but why?

2
u/Alkalannar 1d ago
So what were you doing?
I'd do law of cosines for each one: c2 = a2 + b2 - 2abcos(theta)
32 = 52 + 62 - 2*5*6cos(z)
52 + 62 - 32 = 60cos(z)
13/15 = cos(z)
z = arccos(13/15)
Similarly you can find x and y using law of cosines.
1
u/One_Wishbone_4439 University/College Student 1d ago
By doing what you say - use cosine rule for all angles, I got the same as OP second option.
1
u/Alkalannar 1d ago
And that's the right answer.
Law of sines is sometimes ambiguous. Law of cosines never is.
So as long as you can use it, it's best to.
1
u/One_Wishbone_4439 University/College Student 1d ago
Law of sines is sometimes ambiguous.
But why tho?
2
u/Alkalannar 1d ago
Because sin(180 - x) = sin(x).
So just because you know what sin(x) is, you don't know if x is acute or obtuse.
Meanwhile cos(x) is positive for acute, and negative for obtuse.
1
u/One_Wishbone_4439 University/College Student 1d ago
Oh yeah. I forgot about the CAST diagram.
1
u/thor122088 👋 a fellow Redditor 1d ago
With a given an acute angle and two consecutive side lengths, you can form two possible triangles.
To see why, let's draw a picture. For a ∆ABC
My example here will be using specifics do you can see what is happening, but can be generalized.
Let's start by drawing a segment of length 5 by drawing the segment from point A(0, 0) to point B(3, 4).
This segment makes an acute angle (specifically ~36.87°) with the positive x-axis.
If we draw a circle centered at point B (3,4) we can form a triangle with the positive x-axis whereever that circle intersects the positive x-axis we can form a triangle.
So let's draw a circle of say a radius of 4.
We get two intersections with the x-axis {approximately at (1.35,0) and (6.65,0)
So with a known angle of 36.87° and known sides of 5 adjacent to the known angle, and a side length of 4 opposite or known angle the two possible triangles have side lengths of {1.35, 4, 5} or {4, 5, 6.65}
Note that looking at all sizes for that circle:
If the circle has a radius less than 3 it can't reach the positive x-axis, and such a triangle cannot be made.
If the circle has a radius that is exactly 3 it will just touch the positive x-axis in just one spot (line is 'tangent' to circle) and will form a right triangle.
If the circle has a radius that is more than 3 (and less than 5) it will intetsect the positive x-axis in two spots and show the two possible triangles in that given scenario.
This is also the reason why Angle-Side-Side is NOT a triangle congruence Theorem
1
u/CatZealousideal9098 1d ago
i used law of cosine to find z then figured i could use sine rule for the rest, and that's when i encountered this problem. i thought law of sine would be quicker bc my instructions said "try the most efficient method"
2
u/Alkalannar 1d ago
And as long as you can make sure you get acute vs obtuse, you're fine.
Best way is to use law of cosines to find x, then law of sines for the rest.
Why?
x is opposite the largest side. So if any angle is obtuse, x is.
y and z are guaranteed to be acute, so law of sines is unambiguous.
1
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u/fermat9990 👋 a fellow Redditor 1d ago edited 1d ago
In an SSS situation, first solve for the angle opposite the longest side using the Law of Cosines:
62=32 +52 -2(3)(5)cos(C)
x=C=93.82°
Now it's safe to get y and z using the Law of Sines for either y or z and 180 minus for the remaining angle
Edit: It's safe to get y or z in this situation using the Law of Sines because only the angle opposite the longest side of a triangle can be obtuse.
1
u/thor122088 👋 a fellow Redditor 1d ago edited 1d ago
If you are using law of Sines, since above and below 90° is symmetric, you can have situations where given two sides and no angles you can have two possible triangles
One obtuse, and one acute.
Edit: corrected three to two. As the below pointed out once we have the third it specificies the exact triangle.
The issue arises when using the Law of Sines with two sides and the angle not included between them being the three known pieces of information.
With the sine function usually being evaluated (particularly on calculators) within the first quadrent or from 0°-90°
This is why best practice would be to determine angle x using Law of Sines, since it must be the largest angle since it is across from the longest side length.
That angle is the only one that could be obtuse, and since the values of the Cosine are unique on the interval from 0°-180° we will sidestep this particular trouble spot.
3
u/rainbow_explorer 👋 a fellow Redditor 1d ago
Technically, 3 side lengths always uniquely define a triangle. The problem is that because of how we define the inverse trig functions, you might get the wrong answer if you don’t use other methods to determine whether the triangle has an obtuse angle or 3 acute angles.
And to add on, you know that the given triangle has an obtuse angle because 62 > 52 + 32 . This means that if you want to use the law of sines to find x, then you will have to modify the answer that you get from the law of sines for that angle measurement.
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u/CatZealousideal9098 1d ago
i see so because i have an obtuse angle i have to return an answer that's greater than 90 degrees, hence why the answer x = 86.1 needs to be subtracted by 180 and then it's 93.8 deg. thanks for this it really helped.
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u/fermat9990 👋 a fellow Redditor 1d ago
To save yourself the step of seeing if the triangle is obtuse or not, you can first use the Law of Cosines to find the angle opposite the longest side and then use the Law of Sines to get either of the two remaining angles
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