2. Factor and find all the zeros: b. p 6 + 7p 3 30 = Identify the domain: 4. Simplify:

Transcription

1 1. Divide: 5x 5 3x 3 + 2x 2 8x + 1 by x Fator and find all the zeros: a. x 3 + 5x 2 3x 15 = 0 b. p 6 + 7p 3 30 = 0 3. Identify the domain: a. f x = 3x 5x 2x Simplify: a. 3x2 +6x+3 3x+3 b. g x = 2 9x 2 1 b. 15x 2 x 4 y 5 9x 5 xy 0

2 The Law of SINES For any triangle (right, aute or obtuse), you may use the following formula to solve for missing sides or angles: a sin b sin sin

3 Use Law of SINES when... You have 3 dimensions of a triangle and you need to find the other 3 dimensions - they annot be just NY 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: S - 2 angles and 1 adjaent side S - 2 angles and their inluded side SS (this is an ambiguous ase)

4 Example 1 You are given a triangle,, with angle = 70, angle = 80 and side a = 12 m. Find the measures of angle and sides b and.

5 Example 1 (on t) b a = 12 The angles in a total 180, so angle = 30. Set up the Law of Sines to find side b: 12 sin 70 b sin sin 80 b sin 70 b 12 sin80 sin m

6 Example 1 (on t) 80 a = 12 Set up the Law of Sines to find side : 12 sin 70 sin b = sin 30 sin70 12 sin 30 sin70 6.4m

7 Example 1 (solution) 80 a = 12 ngle = 30 Side b = 12.6 m Side = 6.4 m 70 b =

8 Example 2 You are given a triangle,, with angle = 115, angle = 30 and side a = 30 m. Find the measures of angle and sides b and.

9 Example 2 (on t) 30 To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. a = b We MUST find angle first beause the only side given is side a. The angles in a total 180, so angle = 35.

10 Example 2 (on t) Set up the Law of Sines to find side b: sin35 b sin 30 a = sin 30 b sin b 35 b 30 sin30 sin m

11 Example 2 (on t) Set up the Law of Sines to find side : sin35 sin115 a = sin115 sin b = sin115 sin m

12 Example 2 (solution) ngle = 35 a = = 47.4 Side b = 26.2 m Side = 47.4 m b = 26.2 Note: Use the Law of Sines whenever you are given 2 angles and one side!

13 The mbiguous ase (SS) When given SS (two sides and an angle that is NOT the inluded 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.

14 The mbiguous ase (SS) In the following examples, the given angle will always be angle and the given sides will be sides a and b. b =? a - we don t know what angle is so we an t draw side a in the right position? =?

15 The mbiguous ase (SS) Situation I: ngle is obtuse If angle is obtuse there are TWO possibilities =? If a b, then a is too short to reah side - a triangle with these dimensions is impossible. =? If a > b, then there is ONE triangle with these dimensions. b a b a? =?? =?

16 The mbiguous ase (SS) Situation I: ngle is obtuse - EXMPLE Given a triangle with angle = 120, side a = 22 m and side b = 15 m, find the other dimensions. 15 = b 120 a = 22 Sine a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines: 22 sin sin 15sin120 22sin 15sin120 sin

17 The mbiguous ase (SS) Situation I: ngle is obtuse - EXMPLE ngle = = 23.8 Use Law of Sines to find side : 15 = b a = sin120 sin 23.8 sin120 22sin sin 23.8 sin m Solution: angle = 36.2, angle = 23.8, side = 10.3 m

18 The mbiguous ase (SS) Situation II: ngle is aute If angle is aute there are SEVERL possibilities. b =? a Side a may or may not be long enough to reah side. We alulate the height of the altitude from angle to side to ompare it with side a.? =?

19 The mbiguous ase (SS) Situation II: ngle is aute =? First, use SOH-H-TO to find h: b h? =? a sin h b h bsin Then, ompare h to sides a and b...

20 The mbiguous ase (SS) Situation II: ngle is aute If a < h, then NO triangle exists with these dimensions. =? b a h =??

21 The mbiguous ase (SS) Situation II: ngle is aute If h < a < b, then TWO triangles exist with these dimensions. b h a b a h If we open side a to the outside of h, angle is aute. If we open side a to the inside of h, angle is obtuse.

22 The mbiguous ase (SS) Situation II: ngle is aute If a > b, then ONE triangle exists with these dimensions. b h a Sine side a is greater than side b, side a annot open to the inside of h, it an only open to the outside, so there is only 1 triangle possible!

23 The mbiguous ase (SS) Situation II: ngle is aute If h = a, then ONE triangle exists with these dimensions. b a = h If a = h, then angle must be a right angle and there is only one possible triangle with these dimensions.

24 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 1 Given a triangle with angle = 40, side a = 12 m and side b = 15 m, find the other dimensions. 15 = b 40 =? a = 12 =?? h Find the height: h bsin h 15sin Sine a > h, but a< b, there are 2 solutions and we must find OTH.

25 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 1 FIRST SOLUTION: ngle is aute - this is the solution you get when you use the Law of Sines! 15 = b 40 h a = sin sin sin 1 15sin sin sin 40 12sin sin 40

26 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 1 SEOND SOLUTION: ngle is obtuse - use the first solution to find this solution. 15 = b 40 a = 12 1st a 1st In the seond set of possible dimensions, angle is obtuse, beause side a is the same in both solutions, the aute solution for angle & the obtuse solution for angle are supplementary. ngle = = 126.5

27 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 1 SEOND SOLUTION: ngle is obtuse 15 = b a = 12 ngle = ngle = = 13.5 sin sin 40 12sin sin 40

28 The mbiguous ase (SS) Situation II: ngle is aute - EX. 1 (Summary) ngle = 53.5 ngle = 86.5 Side = 18.6 ngle = ngle = 13.5 Side = = b 86.5 a = = b a = = = 4.4

29 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 2 Given a triangle with angle = 40, side a = 12 m and side b = 10 m, find the other dimensions. 10 = b 40 =? h a = 12 =?? Sine a > b, and h is less than a, we know this triangle has just ONE possible solution - side a opens to the outside of h.

30 The mbiguous ase (SS) Situation II: ngle is aute - EXMPLE 2 10 = b 40 a = 12 Using the Law of Sines will give us the ONE possible solution: 12 sin sin sin 1 10sin sin sin 40 12sin sin 40

31 The mbiguous ase - Summary if angle is obtuse if angle is aute find the height, h = bsin if a < b no solution if a > b one solution if a < h no solution if h < a < b 2 solutions one with angle aute, one with angle obtuse if a > b 1 solution If a = h 1 solution angle is right (Ex I) (Ex II-1) (Ex II-2)

32 Objetive To use the Law of osines to find unknown parts of a. The Law of osines SS & SSS In, a b 2abos OPP DJ DJ 2 DJ DJ os

33 ? 3 m m SS Law of osines OPP DJ DJ 2 DJ DJ os os m

34 The Law of osines SS & SSS In, os a b 2ab os DJ DJ OPP DJ DJ 1 2

35 Solve the. os os 54.9 os DJ1 DJ 2 OPP 2 DJ DJ Unit ir

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