Law of Sines and Law of Cosines

Definition: Oblique Triangles An oblique triangle is a triangle that has no right angles. C B A a b c To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Oblique Triangles

Solving Oblique Triangles The following cases are considered when solving oblique triangles. Two angles and any side (AAS or ASA) A C c A B c 2. Two sides and an angle opposite one of them (SSA) C c a 3. Three sides (SSS) a c b c a B 4. Two sides and their included angle (SAS) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solving Oblique Triangles

Definition: Law of Sines The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Law of Sines If ABC is an oblique triangle with sides a, b, and c, then C B A b h c a C B A b h c a Acute Triangle Obtuse Triangle Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Law of Sines

Example: Law of Sines - ASA Example (ASA): Find the remaining angle and sides of the triangle. C B A b c 60 10 a = 4.5 ft The third angle in the triangle is A = 180 – C – B = 180 – 10 – 60 = 110. Use the Law of Sines to find side b and c. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Law of Sines - ASA

The Ambiguous Case (SSA) Law of Sines In earlier examples, you saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, then three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles satisfy the conditions.

The Ambiguous Case (SSA)

Example 3 – Single-Solution Case—SSA For the triangle in Figure 6.5, a = 22 inches, b = 12 inches, and A = 42 . Find the remaining side and angles. One solution: a  b Figure 6.5

Now you can determine that Example 3 – Solution cont’d Check for the other “potential angle” C  180  – 21.41  = 158.59 (158.59 + 42 = 200.59 which is more than 180  so there is only one triangle.) Now you can determine that C  180  – 42  – 21.41  = 116.59  Multiply each side by b Substitute for A, a, and b. B is acute.

Multiply each side by sin C. Example 3 – Solution cont’d Then the remaining side is given by Law of Sines Multiply each side by sin C. Substitute for a, A, and C. Simplify.

Example: Single Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 110, a = 125 inches, b = 100 inches C B A b = 100 in c a = 125 in 110 C  180 – 110 – 48.74 = 21.26 Since the “given angle is already obtuse, there will only be one triangle. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Single Solution Case - SSA

Example: No-Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 76, a = 18 inches, b = 20 inches C A B b = 20 in a = 18 in 76 There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: No-Solution Case - SSA

Example: Two-Solution Case - SSA Example (SSA): a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the triangle. A = 58, a = 11.4 cm, b = 12.8 cm C  180 – 58 – 72.2 = 49.8 Check for the other “potential angle” C  180  – 72.2  = 107.8  (107.8  + 58  = 165.8  which is less than 180  so there are two triangles.) Example continues. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Two-Solution Case - SSA

Example: Two-Solution Case – SSA continued Example (SSA) continued: 72.2 10.3 cm 49.8 a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the second triangle. A = 58, a = 11.4 cm, b = 12.8 cm B2  180 – 72.2 = 107.8  C  180 – 58 – 107.8 = 14.2 C A B2 b = 12.8 cm c a = 11.4 cm 58 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Two-Solution Case – SSA continued

Definition: Law of Cosines (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.) Law of Cosines Standard Form Alternative Form Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Law of Cosines

Example: Law of Cosines - SSS Find the three angles of the triangle. C B A 8 6 12 Find the angle opposite the longest side first. Law of Sines: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Law of Cosines - SSS

Example: Law of Cosines - SAS B A 6.2 75 9.5 Solve the triangle. Law of Cosines: Law of Sines: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Law of Cosines - SAS