The Law of Sines and the Law of Cosines



The Law of Sines and the Law of Cosines

Introduction

How do you solve a triangle that is not right triangle? Depending upon what you know about the triangle (angles or sides) you can use the Law of Sines or the Law of Cosines to find the missing sides or angles of any triangle.

Recall SSS, AAS, ASA and SAS. These were postulates from geometry that we used to prove triangles congruent. We are going to use these acronyms to help us determine which law is appropriate to apply to solve a triangle. And, while it was never used to prove triangles congruent, we will also consider the ambiguous case, SSA, where there may be one, two or no triangles to be solved.

Law of Sines

In any [pic] with angles A, B and C opposite sides a, b and c, respectively, the following equation is true:

[pic]

For all of the examples we will use a generic triangle like the one below.

Solving Triangles (AAS, ASA, SAS, SSS, SSA)

AAS

In this case we know two of the angles, [pic], and one side (not included between the two angles).

[pic]

ASA

This time we know the side included between the two known angles.

[pic]

Example:

Given [pic], [pic] and a = 8, solve [pic]

Solution: Make sure your calculator is in degree mode! Sketch a triangle, label the vertices A, B and C and the sides opposite a, b, c. Fill in the known information as shown to the left. Scale doesn’t matter. From the triangle you see you have AAS.

C = 180 – 36 – 48 = 96(

[pic]

[pic]

Law of Cosines

Let [pic]be any triangle with sides and angles labeled in the usual way.

Then:

[pic]

The forms above are useful when solving for a side. If you are using the Law of Cosines to solve for an angle then an alternate form may be more useful.

[pic] (The form is similar for finding [pic].)

SAS

Our given information is two sides and the included angle.

[pic]

We could use Law of Sines or Cosines to find the missing angles, but it is better to use the Law of Cosines since the arccosine function will distinguish between acute and obtuse angles.

[pic] C = 180 – A – B

SSS

We know the three sides of the triangles.

[pic]

[pic]

C = 180 – A – B

Example:

Solve [pic]given a = 11, b = 5 and C = 20(.

Solution: Make sure your calculator is in degree mode! Sketch the triangle and fill in the given information. From the sketch you should see that you have SAS.

[pic]

[pic]

B = 180 – 20 – 34.2 = 125.8(

The Ambiguous Case: SSA

For convenience use [pic] as the angle you know and a and b are the sides you know.

SSA and a > b

Only one (obtuse) triangle is possible.

Use the Law of Sines, [pic], to find B.

C = 180 – A – B

[pic]

SSA and a < b

We could have 0, 1 or 2 triangles.

Calculate [pic]

• SSA and [pic]

No triangle is possible.

• SSA and [pic]

One (right) triangle

C = 180 – 90 – A

[pic]

• SSA and [pic]

TWO! Triangles are possible and both must be found (unless the problem specifies otherwise).

Triangle #1

[pic]

C = 180 – A – B

[pic]

Triangle #2

B’ = 180 – B

C’ = 180 – B’ – A

[pic]

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Use 2nd cos 2nd ANS and let the calculator do the work!

B’ (read as “B prime”) means the alternate or second B used in Triangle #2

Note the primes here, too!

And here! But the given info (A and a) didn’t change.

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