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 ... AAS - 2 angles and 1 adjacent side 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 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 .*

A C B 70° 80° a = 12 c b

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.

A C B 115° 30° a = 30 c b

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. ‘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, NO triangle can be formed 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.

The Ambiguous Case (SSA) Situation I: Angle A is obtuse - EXAMPLE Given a triangle with angle A = 93°, side a = 12 cm and side b = 13 cm, find the other dimensions.

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) 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. A B ? 15 = b C = ? c = ? a = 12 h 40°

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. A B ? 10 = b C = ? c = ? a = 12 h 40°

The Ambiguous Case - Summary if angle A is acute find the height, h = b*sinA 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 right (Ex I) (Ex II-1) (Ex II-2)

The Law of Sines AAS ASA SSA (the ambiguous case) Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.

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