Sin a


When we divide side a by the sine of angle Ait is equal to side b divided by the sine of angle B,and also equal to lớn side c divided by the sine of angle C

Sure ... ?

Well, let"s vị the calculations for a triangle I prepared earlier:


asin A = 8sin(62.2°) = 80.885...

Bạn đang xem: Sin a

= 9.04...

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/aa 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:

asin A = bsin B

We can follow similar steps to lớn include c/sin(C)

How vày 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 lớn rearrange & 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 khổng lồ 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.

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Imagine we know angle A, & sides a & b.

We can swing side a to 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 & an Angle not between" case, & even then not always, but we have khổng lồ 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 bởi 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° & 112.9°:


Both are possible! Each one has the 39° angle, and sides of 41 and 28.

So, always kiểm tra 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.

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So this has only one solution.

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