The Law of Sines (or Sine Rule) is very useful for solving triangles:
a
sin A
=
b
sin B
=
c
sin C
It works for any triangle:
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a, b and c are sides. A, B and C are angles. (Side a faces angle A, |
And it says that:
When we divide side a by the sine of angle A
it is equal to side b divided by the sine of angle B,
and also equal to side c divided by the sine of angle C
Sure … ?
Well, let’s do the calculations for a triangle I prepared earlier:
 |
|
The answers are almost the same!
(They would be exactly the same if we used perfect accuracy).
So now you can see that:
a
sin A
=
b
sin B
=
c
sin C
Is This Magic?
Not really, look at this general triangle and imagine it is two right-angled triangles sharing the side h:
The sine of an angle is the opposite divided by the hypotenuse, so:
| sin(A) = h/b |  |
b sin(A) = h | |
| sin(B) = h/a | |
a sin(B) = h |
a sin(B) and b sin(A) both equal h, so we get:
a sin(B) = b sin(A)
Which can be rearranged to:
a
sin A
=
b
sin B
We can follow similar steps to include c/sin(C)
How Do We Use It?
Let us see an example:
Example: Calculate side “c”
Law of Sines:a/sin A = b/sin B = c/sin C
Put in the values we know:a/sin A = 7/sin(35°) = c/sin(105°)
Ignore a/sin A (not useful to us):7/sin(35°) = c/sin(105°)
Now we use our algebra skills to rearrange and solve:
Swap sides:c/sin(105°) = 7/sin(35°)
Multiply both sides by sin(105°):c = ( 7 / sin(35°) ) × sin(105°)
Calculate:c = ( 7 / 0.574… ) × 0.966…
c = 11.8 (to 1 decimal place)
Finding an Unknown Angle
In the previous example we found an unknown side …
… but we can also use the Law of Sines to find an unknown angle.
In this case it is best to turn the fractions upside down (sin A/a instead of a/sin A, etc):
sin A
a
=
sin B
b
=
sin C
c
Example: Calculate angle B
Start with:sin A / a = sin B / b = sin C / c
Put in the values we know:sin A / a = sin B / 4.7 = sin(63°) / 5.5
Ignore “sin A / a”:sin B / 4.7 = sin(63°) / 5.5
Multiply both sides by 4.7:sin B = (sin(63°)/5.5) × 4.7
Calculate:sin B = 0.7614…
Inverse Sine:B = sin−1(0.7614…)
B = 49.6°
Sometimes There Are Two Answers !
There is one very tricky thing we have to look out for:
Two possible answers.
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Imagine we know angle A, and sides a and b. We can swing side a to left or right and come up with two possible results (a small triangle and a much wider triangle) Both answers are right! |
This only happens in the “Two Sides and
an Angle not between” case, and even then not always, but we have to watch out for it.
Just think “could I swing that side the other way to also make a correct answer?”
Example: Calculate angle R
The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that’s OK. We just use P,Q and R instead of A, B and C in The Law of Sines.
Start with:sin R / r = sin Q / q
Put in the values we know:sin R / 41 = sin(39°)/28
Multiply both sides by 41:sin R = (sin(39°)/28) × 41
Calculate:sin R = 0.9215…
Inverse Sine:R = sin−1(0.9215…)
R = 67.1°
But wait! There’s another angle that also has a sine equal to 0.9215…
The calculator won’t tell you this but sin(112.9°) is also equal to 0.9215…
So, how do we discover the value 112.9°?
Easy … take 67.1° away from 180°, like this:
180° − 67.1° = 112.9°
So there are two possible answers for R: 67.1° and 112.9°:
Both are possible! Each one has the 39° angle, and sides of 41 and 28.
So, always check to see whether the alternative answer makes sense.
- … sometimes it will (like above) and there are two solutions
- … sometimes it won’t (see below) and there is one solution
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We looked at this triangle before. As you can see, you can try swinging the “5.5” line around, but no other solution makes sense. So this has only one solution. |
