Review (Law of sines and cosine) cosines)

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Prudence Griffith
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1 Date:03/7,8/01 Review Objetive: Apply the onept to use the law of the sines and osines to solve oblique triangles Apply the onept to find areas using the law of sines and osines Agenda: Bell ringer voabulary Examples (Power Points Presentation) Class work (ex: 1-7) Closing Ativity: Exit Tiket for students to reflet on Essential Question 54 page 663 and ex: (5,54 page 654 and 43,44 page 66) Homework:

5 The Law of SINES

6 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 A b sin B sin C

7 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 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 - angles and 1 adjaent side ASA - angles and their inluded side SSA (this is an ambiguous ase)

8 Exerise 1 You are given a triangle, ABC, with angle A = 70, angle B = 80 and side a = 1 m. Find the measures of angle C and sides b and. * In this setion, angles are named with apital letters and the side opposite an angle is named with the same lower ase letter.*

9 Exerise 1(on t) AAS angles and 1 adjaent side A 70 B 80 b a = 1 C The angles in a total 180, so angle C = 30. Set up the Law of Sines to find side b: 1 sin 70 b sin 80 1 sin 80 b sin 70 b 1 sin80 sin m

10 Exerise 1(on t) B 80 a = 1 Set up the Law of Sines to find side : 1 sin 70 sin 30 A 70 b = C 1 sin 30 sin70 1 sin 30 sin70 6.4m

11 Exerise 1(solution) B 80 a = 1 Angle C = 30 Side b = 1.6 m Side = 6.4 m A 70 b = C Note: We used the given values of A and a in both alulations. Your answer is more aurate if you do not used rounded values in alulations.

12 Exerise You are given a triangle, ABC, with angle C = 115, angle B = 30 and side a = 30 m. Find the measures of angle A and sides b and.

13 Exerise (on t) ASA B 30 angles and their inluded side To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. a = 30 C 115 b A We MUST find angle A first beause the only side given is side a. The angles in a total 180, so angle A = 35.

14 Exerise (on t) B Set up the Law of Sines to find side b: sin35 b sin 30 a = sin 30 b sin35 C 115 b 35 A b 30 sin30 sin35 6.m

15 Exerise (on t) B Set up the Law of Sines to find side : sin35 sin115 a = sin115 sin35 C b = 6. A 30 sin115 sin m

16 Exerise (solution) B Angle A = 35 a = = 47.4 Side b = 6. m Side = 47.4 m C b = 6. A Note: Use the Law of Sines whenever you are given angles and one side!

17 The Ambiguous Case (SSA) When given SSA (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 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist.

18 The Ambiguous Case (SSA) 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 hange them to simulate the steps provided here. b C =? a - we don t know what angle C is so we an t draw side a in the right position A B? =?

19 The Ambiguous Case (SSA) Situation I: Angle A is obtuse If angle A is obtuse there are TWO possibilities C =? If a b, then a is too short to reah side - a triangle with these dimensions is impossible. C =? If a > b, then there is ONE triangle with these dimensions. b a b a A B? =? A B? =?

20 The Ambiguous Case (SSA) C Situation I: Angle A is obtuse - Exerise 3 Given a triangle with angle A = 10, side a = m and side b = 15 m, find the other dimensions. 15 = b A 10 a = Sine a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines: B sin10 15 sin B 15sin10 sin B B sin 1 15sin10 36.

21 The Ambiguous Case (SSA) Situation I: Angle A is obtuse - Exerise 3 C Angle C = = 3.8 Use Law of Sines to find side : 15 = b A 10 a = B 36. sin10 sin 3.8 sin10 sin 3.8 sin 3.8 sin m Solution: angle B = 36., angle C = 3.8, side = 10.3 m

22 The Ambiguous Case (SSA) Situation II: Angle A is aute If angle A is aute there are SEVERAL possibilities. b C =? a Side a may or may not be long enough to reah side. We alulate the height of the altitude from angle C to side to ompare it with side a. A B? =?

23 The Ambiguous Case (SSA) Situation II: Angle A is aute C =? First, use SOH-CAH-TOA to find h: b h A B? =? a sin A h b h bsin A Then, ompare h to sides a and b...

24 The Ambiguous Case (SSA) Situation II: Angle A is aute If a < h, then NO triangle exists with these dimensions. C =? b a h A =? B?

25 The Ambiguous Case (SSA) Situation II: Angle A is aute If h < a < b, then TWO triangles exist with these dimensions. C C b h a b a h A B A B If we open side a to the outside of h, angle B is aute. If we open side a to the inside of h, angle B is obtuse.

26 The Ambiguous Case (SSA) Situation II: Angle A is aute If h < b < a, then ONE triangle exists with these dimensions. A b C h a B 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!

27 The Ambiguous Case (SSA) Situation II: Angle A is aute If h = a, then ONE triangle exists with these dimensions. A b C a = h B If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions.

28 The Ambiguous Case (SSA) Situation II: Angle A is aute - Exerise 4 Given a triangle with angle A = 40, side a = 1 m and side b = 15 m, find the other dimensions. 15 = b 40 C =? a = 1 A =? B? h Find the height: h bsin A h 15sin Sine a > h, but a< b, there are solutions and we must find BOTH.

29 The Ambiguous Case (SSA) Situation II: Angle A is aute - Exerise 4 FIRST SOLUTION: Angle B is aute - this is the solution you get when you use the Law of Sines! A 15 = b 40 C h a = 1 B 1 sin sin B B sin 1 15sin C sin sin 40 1sin sin 40

30 The Ambiguous Case (SSA) A Situation II: Angle A is aute - Exerise 4 SECOND SOLUTION: Angle B is obtuse - use the first solution to find this solution. 15 = b 40 B C a = 1 1st a 1st B In the seond set of possible dimensions, angle B is obtuse, beause side a is the same in both solutions, the aute solution for angle B & the obtuse solution for angle B are supplementary. Angle B = = 16.5

31 The Ambiguous Case (SSA) Situation II: Angle A is aute - Exerise 4 SECOND SOLUTION: Angle B is obtuse A 15 = b B C a = 1 Angle B = 16.5 Angle C = = 13.5 sin sin 40 1sin sin 40

32 The Ambiguous Case (SSA) Situation II: Angle A is aute - Exerise 4(Summary) Angle B = 53.5 Angle C = 86.5 Side = 18.6 C Angle B = 16.5 Angle C = 13.5 Side = C 15 = b 86.5 a = 1 15 = b a = 1 A = 18.6 B A = 4.4 B

33 The Ambiguous Case (SSA) Situation II: Angle A is aute Exerise 5 Given a triangle with angle A = 40, side a = 1 m and side b = 10 m, find the other dimensions. 10 = b 40 C =? h a = 1 A =? B? 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.

34 Review (Law of sines and osine) The Ambiguous Case (SSA) Situation II: Angle A is aute - Exerise 5 A 10 = b 40 C a = 1 B Using the Law of Sines will give us the ONE possible solution: 1 sin sin B 10sin 40 B sin C sin sin 40 1sin sin 40

35 The Law of Sines a sin A b sin B sin C Use the Law of Sines to find the missing dimensions of a triangle when given any ombination of these dimensions. AAS ASA SSA (the ambiguous ase)

36 Review (Law of sines and osine) The Law of Cosines

37 Review (Law of sines and osines) Let's onsider types of triangles with the three piees of information shown below. We an't use the Law of Sines on these beause we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. SAS You may have a side, an angle, and then another side AAA You may have all three angles. SSS You may have all three sides AAA This ase doesn't determine a triangle beause similar triangles have the same angles and shape but "blown up" or "shrunk down"

38 Review (Law of sines and osines) We ould do the same thing if gamma was obtuse and we ould repeat this proess for eah of the other sides. We end up with the following: b LAW OF COSINES a a a b b Use these to find missing angles abos aos bos os b LAW OF COSINES a b a os Use these to find missing sides os b ab a b a

39 Review (Law of sines and osines) Exerise 6 Solve a triangle where b = 1, = 3 and = a 1

40 CAUTION: Don't forget order of operations: powers then multipliation BEFORE addition and subtration Review (Law of sines and osines) Solve a triangle where b = 1, = 3 and = 80 Draw a piture. This is SAS Do we know an angle and side opposite it? No so we must use Law of Cosines. Hint: we will be solving for the side opposite the angle we know. One side squared Now punh buttons on your alulator to find a. It will be square root of right hand side. a =.99 a a b 80 3 sum of eah of the other sides squared b a minus times the produt of those other sides 3 os times the osine of the angle between those sides 1 os 80

41 Review (Law of sines and osines) We'll label side a with the value we found. We now have all of the sides but how an we find an angle? Hint: We have an angle and a side opposite it. sin80 sin sin is easy to find sine the sum of the angles is a triangle is NOTE: These answers are orret to deimal plaes for sides and 1 for angles. They may differ with book slightly due to rounding. Keep the answer for in your alulator and use that for better auray.

42 Review (Law of sines and osines) Exerise 7 Solve a triangle where a = 5, b = 8 and =

43 Review (Law of sines and osines) Solve a triangle where a = 5, b = 8 and = 9 Draw a piture. This is SSS Do we know an angle and side opposite it? No, so we must use Law of Cosines. Let's use largest side to find largest angle first. One side squared os 8 os 1 os a b sum of eah of the other sides squared ab os minus times the produt of those other sides 5 8 CAUTION: Don't forget order of operations: powers then multipliation BEFORE addition and subtration times the osine of the angle between those sides os

44 Review (Law of sines and osines) How an we find one of the remaining angles? Do we know an angle and side opposite it? Yes, so use Law of Sines. sin 84.3 sin 9 8 8sin sin sin 84.3 sin 6. 9

Solving an Oblique Triangle

Solving an Oblique Triangle Several methods exist to solve an oblique triangle, i.e., a triangle with no right angle. The appropriate method depends on the information available for the triangle. All methods require that the length