SohCahToa hypotenuse opposite sin hypotenuse adjacent cos adjacent opposite tan Soh C ah Toa
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.