When we divide side a by the sine of angle Ait is equal lớn 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 vì chưng the calculations for a triangle I prepared earlier:
 | asin A = 8sin(62.2°) = 80.885... bsin B = 5sin(33.5°) = 50.552... = 9.06... csin C = 9sin(84.3°) = 90.995... = 9.04... |
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 & 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) và b sin(A) both equal h, so we get:
a sin(B) = b sin(A)
Which can be rearranged to:
asin A = bsin B
We can follow similar steps to include c/sin(C)
How bởi vì 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 lớn us):7/sin(35°) = c/sin(105°)
Now we use our algebra skills khổng lồ 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 khổng lồ find an unknown angle.
In this case it is best lớn 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 lớn look out for:
Two possible answers.
 | Imagine we know angle A, và sides a và b. We can swing side a to lớn left or right & come up with two possible results (a small triangle và a much wider triangle) Both answers are right! |
This only happens in the "Two Sides và an Angle not between" case, & even then not always, but we have to lớn watch out for it.
Just think "could I swing that side the other way to lớn also make a correct answer?"
Example: Calculate angle R
The first thing lớn 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 và 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 lớn 0.9215...
The calculator won"t tell you this but sin(112.9°) is also equal lớn 0.9215...
So, how vày 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° và 112.9°:
Both are possible! Each one has the 39° angle, & sides of 41 và 28.
So, always check to see whether the alternative answer makes sense.
... Sometimes it will (like above) & there are two solutions... Sometimes it won"t (see below) và there is one solution | 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. |
The Law of Cosines Solving Triangles Trigonometry Index Algebra Index