The Law of SINES. For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

Transcription

1 The Law of SINES

2 The Law of SINES For any triangle (right, aute or otuse), you may use the following formula to solve for missing sides or angles: a sin = 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 e 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 amiguous 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 and. * In this setion, angles are named with apital letters and the side opposite an angle is named with the same lower ase letter.*

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

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

7 Example 1 (solution) ngle = 30 = a = 12 Side = 12.6 m Side = 6.4 m 70 = Note: We used the given values of and a in oth alulations. Your answer is more aurate if you do not used rounded values in alulations.

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 and.

9 Example 2 (on( 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 = We MUST find angle first eause the only side given is side a. The angles in a total 180, so angle = 35.

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

11 Example 2 (on( on t) Set up the Law of Sines to find side : sin35 = sin115 a = sin115 = sin = 26.2 = 30 sin115 sin m

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

13 The miguous ase (SS) When given SS (two sides and an angle that is NOT the inluded angle), the situation is amiguous. The dimensions may not form a triangle, or there may e 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 miguous ase (SS) In the following examples, the given angle will always e angle and the given sides will e sides a and. If you are given a different set of variales, feel free to hange them to simulate the steps provided here. =? a - we don t know what angle is so we an t draw side a in the right position? =?

15 The miguous ase (SS) Situation I: ngle is otuse If angle is otuse there are TWO possiilities =? If a, then a is too short to reah side - a triangle with these dimensions is impossile. =? If a >, then there is ONE triangle with these dimensions. a a? =?? =?

16 The miguous ase (SS) Situation I: ngle is otuse - EXMPLE Given a triangle with angle = 120, side a = 22 m and side = 15 m, find the other dimensions. 15 = 120 a = 22 Sine a >, these dimensions are possile. To find the missing dimensions, use the Law of Sines: 22 sin120 = 15 sin 15sin120 = 22sin = sin 1 15sin = 36.2

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

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

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

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

21 The miguous ase (SS) Situation II: ngle is aute If h < a <, then TWO triangles exist with these dimensions. h a 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 otuse.

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

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

24 The miguous ase (SS) Situation II: ngle is aute - EXMPLE 1 Given a triangle with angle = 40, side a = 12 m and side = 15 m, find the other dimensions. 15 = 40 =? a = 12 =?? h Find the height: h = sin h = 15sin40 = 9.6 Sine a > h, ut a<, there are 2 solutions and we must find OTH.

25 The miguous 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 = 40 h a = sin 40 = 15 sin = sin 1 15sin40 12 = 53.5 = = 86.5 sin86.5 = 12 sin 40 = 12sin86.5 = 18.6 sin 40

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

27 The miguous ase (SS) Situation II: ngle is aute - EXMPLE 1 SEOND SOLUTION: ngle is otuse 15 = a = 12 ngle = ngle = = 13.5 sin13.5 = 12 sin 40 = 12sin13.5 = 4.4 sin 40

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

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

30 The miguous ase (SS) Situation II: ngle is aute - EXMPLE 2 10 = 40 a = 12 Using the Law of Sines will give us the ONE possile solution: 12 sin 40 = 10 sin = sin 1 10sin = 32.4 = =107.6 sin107.6 = 12 sin 40 = 12sin107.6 = 17.8 sin 40

31 The miguous ase - Summary if angle is otuse if angle is aute find the height, h = *sin if a < no solution if a > one solution if a < hno solution if h < a < 2 solutions(ex II-1) one with angle aute, one with angle otuse if a > > h1 solution (Ex II-2) If a = h1 solution angle is right (Ex I)

32 The Law of Sines a sin = sin = sin Use the Law of Sines to find the missing dimensions of a triangle when given any omination of these dimensions. S S SS (the amiguous ase)

33 dditional Resoures We Links: