The Law of SINES The Law of SINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles: a b c sin A sin B sin C Use Law of SINES when ... 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 wont 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

ASA - 2 angles and their included side SSA (this is an ambiguous case) Example 1 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 .* Example 1 (cont) B The angles in a total 180, so angle C = 30. 80 a = 12

c A 70 b Set up the Law of Sines to find side b: C 12 b sin 70 sin 80 12 sin 80 b sin 70 12 sin80 b 12.6cm sin 70 Example 1 (cont) B 80

c A 70 Set up the Law of Sines to find side c: a = 12 b = 12.6 30 12 c sin 70 sin 30 C 12 sin 30 c sin70 12 sin 30 c 6.4cm sin70

B Angle C = 30 80 Side b = 12.6 cm a = 12 Side c = 6.4 cm c= 6 .4 Example 1 (solution) A 70 b = 12.6

30 Note: C We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations. Example 2 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. Example 2 (cont) To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation.

B 30 c a = 30 115 C b We MUST find angle A first because the only side given is side a. A The angles in a total 180, so angle A = 35. Example 2 (cont) B Set up the Law of Sines to find side b: 30 b

sin35 sin 30 30 c a = 30 115 35 C b 30 sin 30 b sin35 A 30 sin30 b 26.2cm sin35 Example 2 (cont) B Set up the Law of Sines to find side c: 30

c a = 30 115 35 C b = 26.2 A 30 c sin35 sin115 30 sin115 c sin35 30 sin115 c 47.4cm sin35 Example 2 (solution) B Angle A = 35 30

Side b = 26.2 cm c = 47.4 a = 30 115 35 C b = 26.2 A Side c = 47.4 cm Note: Use the Law of Sines whenever you are given 2 angles and one side! The Ambiguous Case (SSA) 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.

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 change them to simulate the steps provided here. C=? angle C is so we cant draw side a in the right position b A a - we dont know what c=? B? The Ambiguous Case (SSA) 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 C = ? impossible. a b A C=? If a > b, then there is ONE triangle with these dimensions. a b c=? B? A c=?

B? The Ambiguous Case (SSA) 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. Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines: C 22 15 sin120 sin B 15sin120 22sin B a = 22 15 = b

A 120 c B 15sin120 B sin 1 36.2 22 The Ambiguous Case (SSA) Situation I: Angle A is obtuse - EXAMPLE Angle C = 180 - 120 - 36.2 = 23.8 C Use Law of Sines to find side c: a = 22

15 = b A 120 c B 36.2 22 c sin120 sin 23.8 c sin120 22sin 23.8 22sin 23.8 c 10.3cm sin120 Solution: angle B = 36.2, angle C = 23.8, side c = 10.3 cm

The Ambiguous Case (SSA) Situation II: Angle A is acute If angle A is acute there are SEVERAL possibilities. C=? b A a c=? 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. B? The Ambiguous Case (SSA) Situation II: Angle A is acute First, use SOH-CAH-TOA to find h: C=? b

a h A c=? B? h sin A b h bsin A Then, compare h to sides a and b . . . The Ambiguous Case (SSA) Situation II: Angle A is acute If a < h, then NO triangle exists with these dimensions. C=? a

b h A c=? B? The Ambiguous Case (SSA) Situation II: Angle A is acute If h < a < b, then TWO triangles exist with these dimensions. C b h A c C b a B

If we open side a to the outside of h, angle B is acute. A c a h B If we open side a to the inside of h, angle B is obtuse. The Ambiguous Case (SSA) Situation II: Angle A is acute If h < b < a, then ONE triangle exists with these dimensions. C b a h

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

c B If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions. The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 1 Given a triangle with angle A = 40, side a = 12 cm and side b = 15 cm, find the other dimensions. Find the height: C=? h bsin A h 15sin40 9.6 a = 12 15 = b

h A 40 c=? B? Since a > h, but a< b, there are 2 solutions and we must find BOTH. The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 1 FIRST SOLUTION: Angle B is acute - this is the solution you get when you use the Law of Sines! C a = 12 15 = b

h A 40 c B 12 15 sin 40 sin B 15sin40 B sin 1 53.5 12 C 180 40 53.5 86.5 c 12 sin86.5 sin 40

12sin86.5 c 18.6 sin 40 The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 1 SECOND SOLUTION: Angle B is obtuse - use the first solution to find this solution. C 1st a 15 = b A a = 12 40 c B

1st B In the second set of possible dimensions, angle B is obtuse, because side a is the same in both solutions, the acute solution for angle B & the obtuse solution for angle B are supplementary. Angle B = 180 - 53.5 = 126.5 The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 1 SECOND SOLUTION: Angle B is obtuse C Angle B = 126.5 Angle C = 180- 40- 126.5 = 13.5 15 = b a = 12 A

40 126.5 c B c 12 sin13.5 sin 40 12sin13.5 c 4.4 sin 40 The Ambiguous Case (SSA) Situation II: Angle A is acute - EX. 1 (Summary) Angle B = 126.5 Angle C = 13.5 Side c = 4.4 Angle B = 53.5 Angle C = 86.5

Side c = 18.6 13.5 C 15 = b A 40 86.5 15 = b a = 12 53.5 c = 18.6 B C

a = 12 A 40 126.5 B c = 4.4 The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 2 Given a triangle with angle A = 40, side a = 12 cm and side b = 10 cm, find the other dimensions. C=? a = 12 10 = b h A 40

c=? B? Since a > b, and h is less than a, we know this triangle has just ONE possible solution - side aopens to the outside of h. The Ambiguous Case (SSA) Situation II: Angle A is acute - EXAMPLE 2 Using the Law of Sines will give us the ONE possible solution: C a = 12 10 = b A 40

c B 12 10 sin 40 sin B 10sin 40 1 B sin 32.4 12 C 180 40 32.4 107.6 c 12 sin107.6 sin 40 12sin107.6 c 17.8

sin 40 The Ambiguous Case Summary if angle A is obtuse if a < b no solution if a > b one solution (Ex I) if a < h no solution if angle A is acute find the height, h = b*sinA if h < a < b 2 solutions(Ex II-1) one with angle B acute, one with angle B obtuse if a > b > h 1 solution (Ex II-2) If a = h 1 solution angle B is right

The Law of Sines a b c sin A sin B sin C 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) Additional Resources

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