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 ASA SSA (the 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 (con’t) A C B 70° 80° a = 12 c b The angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b:

Example 1 (con’t) A C B 70° 80° a = 12 c b = 12.6 30° Set up the Law of Sines to find side c:

Example 1 (solution) A C B 70° 80° a = 12 c = 6.4 b = 12.6 30° Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Note: 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. We MUST find angle A because the only side given is side a. The angles in a ∆ total 180°, so angle A = 35°. A C B 115° 30° a = 30 c b

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

Example 2 (con’t) A C B 115° a = 30 c b = 26.2 30° a = 30 c b = 26.2 35° Set up the Law of Sines to find side c:

Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm A 115° 30° a = 30 c = 47.4 b = 26.2 35° Angle A = 35° Side b = 26.2 cm 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. ‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position A B ? b C = ? c = ?

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 impossible. If a > b, then there is ONE triangle with these dimensions. A B ? a b C = ? c = ? A B ? a b C = ? c = ?

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: A B a = 22 b = 15 C c 120°

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

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

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

The Ambiguous Case (SSA) Situation II: Angle A is acute If h = a, then ONE triangle exists with these dimensions. A B b C c a = h 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 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 if angle A is acute find the height, h = b*sinA

The Law of COSINES

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