Math-3 Lesson 6-5 The Law of Sines The miguous Case
Quiz 6-4: 1. Find the measure of angle θ. Ө = 33.7 2. What is the cosecant ratio for ϴ? Csc Ө = 2 5 5 3. standard position angle passes through the point (8, -3). Sin Ө =? 2 3 5 Sin Ө = -3 73 73 4. Solve for x using the Law of Sines. x = 33.9 a sin x 25 sin B 15 107 c sin C
What is the pattern for Law of Sines? C a If three of the four circled numers are known, use law of sines. c B sin sin B a or a sin sin B
Your Turn: = 75º B = 20º = 20 Given the following: solve the triangle. 1. Draw and lael the triangle. 2. a =? 3. C =? 4. c =?
Do you rememer the 4 triangle congruence theorems? C E 48 15 15 52 52 48 B D F ngle, (included) side, angle S
Triangle Review C If the following information is given: Two angles and a non-included side 15 ngle, angle, side S 25 c 107 Can you use the Law of Sines for S?
Triangle Review C If the following information is given: Two ngles and the included side. 48 a ngle, side, angle S 25 15 107 Can you use the Law of Sines for S? If you have two angles of a triangle, you can find the 3 rd angle.
Triangle Review C If the following information is given: Two sides and the included angle. 12 a Side, ngle, Side SS 25 15 Law of Sines will NOT work for SS.
Triangle Review C If the following information is given: Three sides. 12 11 Side, Side, Side SSS 15 B Law of Sines will NOT work for SSS.
12 25 15 B If the following information is given: n angle and its opposite side and one other side. Side, Side, ngle SS Will the Law of Sines work for SS? It depends
Your Turn: Can the Law of sines e used for: SS? no SSS? S?? no yes, BUT. no, trangle can e scaled up or down in size (no unique triangle).
Triangle Review: C a When any comination of three known values (sides/angles) are are given, the possile cases are: S S c B SS (the polite order of its letters) this one presents some prolems for the Law of Sines. SS: Can t use Law of Sines. (can e any size, can t uild a specific triangle) SSS: (has no angle given, can t use Law of Sines)
SS: The miguous Case If angle and opposite side a, and one other side are given: (1) If the adjacent side is longer than the opposite side ND (2) If the given angle is acute I can swing side a until it touches the ottom side at its end point. This makes another triangle. 15 10 10 25 Two triangles are possile.
SS: The non- miguous Case If angle and opposite side a, and one other side are given: (1) If the adjacent side is shorter than the opposite side 10 15 25 Only one triangle is possile.
SS: The non- miguous Case If angle and opposite side a, and one other side are given: (1) If the angle is otuse 10 Only one triangle is possile. 125 15 NOTICE: the opposite side must e longer than the adjacent side. Why?
SS Case If the given information aout a triangle is: = 68º = 88 a = 85 1. Draw the triangle. The angle is not etween 2. Lael the triangle the two sides. 3. Determine what case it is. C This may cause prolems. 88 a85 We call this the amiguous case. 68º c B miguous: Liale to more than one interpretation.
What is a triangle? 3 segments joined at their endpoints
SS: The miguous Case IF: the side opposite the given angle is shorter than the adjacent side, you may have two triangles. 5 25 4 25 5 4
Which of the following cases might give you two possile triangles? = 68º = 68 a = 85 = 25º = 7 a = 5 = 118º = 8 a = 20 IF: (1) The given angle is acute (<90 ) and (2) the side opposite the given angle is shorter than the adjacent side, you may have two triangles.
IF: (1) angle is acute (<90 ) (2) opposite < adjacent side, there are three possiilities ( cases ). 1. The opposite side is too short to even make a triangle. 2. The opposite side is just long enough to touch right triangle. a a 3. The opposite side can touch in two places 2 triangles. = 25º = 7 a = 5 a a
How can you tell which case it is? a = 25º = 7 a = 5 Use the non-opposite side and the angle to determine the side length that would give a right triangle. a How would this help you to find which case it is? a a
SS Case: djacent > opposite Find the opposite side length to make a right triangle. 7 5 y 25 y sin 25 7 y = 7 sin 25º y = 2.95 y = opposite side length for a right triangle one triangle. = 25º, = 7, a = 5 To make a right angle: a must equal 2.95. 5 > just right length. 2 triangles.
= 36.9º, = 5, a = 4 How many triangles are possile? 5 4 y 36.9 y sin 36.9 5 y = 5 sin 36.9º y = opposite side length for a right triangle one triangle 4 > just right length. 2 triangles. y = 3
SS Case Side opposite the angle is shorter than the shortest possile opposite side. No triangle!!! 5 36.9º 3 C 2 B
Summary (amiguous case) adj angle opp s & S s S SS SS Given ngle and Sides dj > opp; opp > just right dj > opp; opp = just right dj > opp; opp < just right djacent side < opposite side #of Triangles: 2 1 0 1
How many triangles? (one, two, or none) = 52º, a = 32, = 42 = 28º, C = 75º, c = 20 = 40º, a = 13, = 16 (Hint: draw the triangle!!!)
Solve the Triangle: sin 20 y 5 Opp side > right angle side? = 20º, a = 4, = 5 1) Draw the triangle (amiguous SS Case?) (adj > opp) YES! 2) How many triangles? find opposite side for a right triangle. y 5sin 20 1. 71 2 triangles 5 y 4 20 B
Solve the Triangle: If it is two triangles, the law of sines will give the acute angle: 5 = 20º, a = 4, = 5 20 20 sin 1 5 sin 1 2 sin 20 4 5sin 20 4 4 sin 1 0.4275 1 4 1 sin 1 1 25.3 5 0.4275 m C 180 20 25.3 C 134.7 c Use law of sines to find c. 1 4 134.7 B
Note: Do you see the Blue Isosceles Triangle? 20 20 5 4 c 2 25.3 5 4 154.7 4 25.3 C B Both ase angles are congruent. Note: Do you see the linear pair of angles? 180 25.3 = 154.7 Solve the 2 nd triangle. m C 180 20 154.7 = 5.3 Use Law of Sines to solve for c. 2
Finding the Height of a Pole Two people are 2.5 meters apart on opposite sides of a pole. The angles of elevation from the oservers to the top of the pole are 51 and 68. Find the height of the pole. a 51º 68º 2.5 1. Find either a or using Law of Sines. 2. Solve the right triangle using right triangle rules where height is the side opposite the angle.