The Law of SINES The Law of SINES
- Slides: 31
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:
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 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 l ASA - 2 angles and their included side l SSA (this is an ambiguous case) l
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 (AAS) 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
Example 1 (AAS) B 80° c A 70° Set up the Law of Sines to find side c: a = 12 b = 12. 6 30° C
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 (ASA) 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 (ASA) Set up the Law of Sines to find side b: B 30° c a = 30 115° 35° C b A
Example 2 (ASA) Set up the Law of Sines to find side c: B 30° c a = 30 115° 35° C b = 26. 2 A
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 can’t draw side ‘a’ in the right position b A ‘a’ - we don’t know what c=? B?
The Ambiguous Case (SSA) Situation I: Angle A is obtuse If angle A is obtuse, then side a MUST be the longest side. If a ≤ b, then a is too short to reach side c – this triangle is impossible. C=? 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 a = 22 15 = b A 120° c B
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° 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. If a < h, then NO triangle exists with these dimensions. C=? Use SOH-CAH-TOA to find h: a b h A c=? B? Then compare a to h to ensure you have a triangle. If not, a solution is impossible.
The Ambiguous Case (SSA) Situation II: Angle A is acute Once you have confirmed a triangle exists, compare a to b. If h < a < b, then TWO triangles are possible. C C b h A 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 a>b , then ONE triangle is possible. 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 Special Case: If a= h, then you have a right triangle 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=? 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
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 1 st ‘a’ 15 = b A a = 12 40° c B 1 st ‘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
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 ‘a’opens 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
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*sin. A 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 Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions. AAS l ASA l SSA (the l ambiguous case)
Additional Resources l Web Links: http: //www. regentsprep. org/regents/math/algtrig/ATT 1 2/lawofsines. htm l http: //oakroadsystems. com/twt/solving. htm#Sine. Law l http: //oakroadsystems. com/twt/solving. htm#Detective l
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