The Laws of SINES and COSINES The Law

The Laws of SINES and COSINES

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 given. . . AAS l ASA l SSA (the 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 (con’t) 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 (con’t) 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 since your answer is more accurate if you do not use 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 (con’t) To solve for the missing sides/angles, we must have an angle/side opposite pair to set up the first equation. B 30° c a = 30 115° C b We MUST find angle A because the only side given is side a. A The angles in a ∆ total 180°, so angle A = 35°.

Example 2 (con’t) Set up the Law of Sines to find side b: B 30° c a = 30 115° 35° C b A

Example 2 (con’t) 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 those 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 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. C To find the missing dimensions, use the Law of Sines: a = 22 b = 15 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 b = 15 A 120° c B 36. 2° Solution: angle B = 36. 2°, angle C = 23. 8°, side c = 10. 3

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? 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) if angle A is obtuse if a < b no solution if a > b one solution if a < h no solution if angle A is acute find the height, h = b*sin. A if h < a < b 2 solutions one with angle B acute, one with angle B obtuse if a > b > h 1 solution If a = h 1 solution angle B is a right angle

The Law of COSINES

Use Law of COSINES when given. . . SAS l SSS (start with the l largest angle!)