Chapter 6

 Use the law of sines to solve triangles

 For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

 you have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given:  AAS - 2 angles and 1 adjacent side or ASA - 2 angles and their included side  SSS- three sides  SSA (this is an ambiguous case)

You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c.  * In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter.*

  The angles in a ∆ total 180°, so angle C = 30°.  Set up the Law of Sines to find side b: AC B 70° 80° a = 12 c b

 B:

 You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c.

AC B 115° 30° a = 30 c b

 For the triangle in fig 6.3 C=102.3 degree, B=28.7 degree and b=27.4 feet  C   A B 

 When given SSA (two sides and an angle that is NOT the included angle), the situation is ambiguous. The dimensions may not form a triangle, or there may be 1 or 2 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist.

 In the following examples, the given angle will always be angle A and the given sides will be sides a and b. If you are given a different set of variables, feel free to change them to simulate the steps provided here. ‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position AB ? b C = ? c = ?

 Situation I: Angle A is obtuse  If angle A is obtuse there are TWO possibilities  If a ≤ b, then a is too short to reach side c - a triangle with these dimensions is impossible. AB ? a b C = ? c = ?

 If a > b, then there is ONE triangle with these dimensions. AB ? a b C = ? c = ?

 Situation I: Angle A is obtuse - EXAMPLE  Given a triangle with angle A = 120°, side a = 22 cm and side b = 15 cm, find the other dimensions. A B a = = b C c 120° Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines:

Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3 cm

 Situation II: Angle A is acute  If angle A is acute there are SEVERAL possibilities.  Side ‘a’ may or may not be long enough to reach side ‘c’. We calculate the height of the altitude from angle C to side c to compare it with side a. AB ? b C = ? c = ? a

 Situation II: Angle A is acute  First, use SOH-CAH-TOA to find h: AB ? b C = ? c = ? a h Then, compare ‘h’ to sides a and b...

 If a < h, then NO triangle exists with these dimensions. AB ? b C = ? c = ? a h

 If h < a < b, then TWO triangles exist with these dimensions. AB b C c a h If we open side ‘a’ to the outside of h, angle B is acute A B b C c a h If we open side ‘a’ to the inside of h, angle B is obtuse.

 If h < b < a, then ONE triangle exists with these dimensions. AB b C c a h Since side a is greater than side b, side a cannot open to the inside of h, it can only open to the outside, so there is only 1 triangle possible!

 If h = a, then ONE triangle exists with these dimensions. A B b C c a = h If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions.

 Given a triangle with angle A = 40°, side a = 12 cm and side b = 15 cm, find the other dimensions. A B ? 15 = b C = ? c = ? a = 12 h 40°

 Angle B = 53.5°  Angle C = 86.5°  Side c = 18.6  Angle B = 126.5°  Angle C = 13.5°  Side c = 4.4

if angle A is acute find the height, h = b*sinA if angle A is obtuse if a < b  no solution if a > b  one solution if a < h  no solution if h < a < b  2 solutions one with angle B acute, one with angle B obtuse if a > b > h  1 solution If a = h  1 solution angle B is right (Ex I) (Ex II-1) (Ex II-2)

 Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.  AAS  ASA  SSA (the ambiguous case)

 Do problems 7-12 in your book page 410 

 Do problems 13-17, in your book page 410

 Today we learned about the law of sines  Next class we are going to learn about law of cosines