SohCahToa

hypotenuseoppositesin

hypotenuseadjacentcos

adjacentoppositetan

Soh

Cah

Toa

Trigonometric Ratios III

Objectives:

1. To derive the Laws of Sines and Cosines

2. To find all the angles and sides of any triangle using the Laws of Sines and Cosines

Investigation: Law of Sines

Trigonometry can be applied to non-right, or oblique, triangles. In that example, we used it to find an unknown height. Now, we’ll use it to find a missing side length of a non-right triangle.

Law of Sines

If ΔABC has side lengths a, b, and c as shown, then

sin sin sinA B C

a b c

Looking for an ANGLE:

sin sin sin

a b c

A B C

Looking for a SIDE:

Example 2

Find the length of side AC in ΔABC.

Example 3

Find the length of side AC in ΔABC.

Acute or Obtuse?

The Law of Sines can also be used to find a missing angle measure, but only if you know that it is acute or obtuse. This is simply because SSA is not a congruence shortcut.

160260

36

B1

C

A

160

260

36B2

C

A

Example 4

Solve ΔABC.

Example 5

Find the indicated measure.1. 2. and and x y

Example 6

Find the length of AC in acute triangle ABC.

Law of Sines

As the previous example demonstrates, you cannot always use the Law of Sines for every triangle. You need either two sides and an angle or two angles and a side in the following configurations:

ASA

AAS

SSA

Law of Cosines

If ABC has sides of length a, b, and c as shown, then

Abccba

Baccab

Cabbac

cos2

cos2

cos2

222

222

222

Example 7

Simplify the equation below for C = 90°.2 2 2 2 cosc a b ab C

Example 8

Solve the equation below for C.2 2 2 2 cosc a b ab C

Example 9

Find the length of AC in acute triangle ABC.

Example 10

Solve ΔABC.

Pro Tip: SSS

When using the Law of Cosines to find a missing angle (SSS), it’s a good idea to find the angle opposite the longest side first. This is just in case the angle turns out to be obtuse. Regardless of what type of angle this turns out to be, use the Law of Sines and the Triangle Sum Theorem to find the other two angles.

1: Use the Law of Cosines

2: Use the Law of Sines

3: Use the Triangle Sum

Example 11

Find the indicated measure.1. 2. and and x y

Summary

• ASA• AAS• SSA

Law of Sines

• SAS• SSSLaw of

Cosines

Given three pieces of any triangle, you can use the Law of Sines or the Law of Cosines to completely solve the triangle.