Objectives: 1.Be able to prove the Law of Sines using Right Triangle Trigonometry. 2.Be able to apply the Law of Sines on various triangles. 3.Be able to determine the number of triangles that exist in the ambiguous case (SSA). Critical Vocabulary: Soh-Cah-Toa, Sine Function, Ambiguous Case

Review Problem Find: Height 17 53

   The Law of Sines is three different formulas that help solve triangles that are not right triangles. These formulas do also work on right triangles The formula states the ratio of the sine of an angle and its opposite side is equal to sine of another angle and its opposite side. Take a look at the first of three formulas using angle alpha and angle beta. Take a look at the second of three formulas using angle beta and angle gamma. Take a look at the last formula using angle alpha and angle gamma. Where did these formulas actually come from? On the next slide we will see where these formulas came from. These formulas will help us solve: SAA, SSA, and ASA triangles.

   Let’s drop down an ALTITUDE from one of the vertices An ALTITUDE will form 2 right angles. However, it does not divide the length into two equal parts. Lets label the altitude of the triangle. x a b c Let’s look at the right triangle on the left. Using the alpha angle find the sine (opposite/hypotenuse) Let’s look at the right triangle on the right. Using the gamma angle find the sine (opposite/hypotenuse) Set the two equations equal since they both equal “x” Divide both sides by “a” Divide both sides by “c”Now, Prove the other 2/3 of the formula. Redraw the triangle and move the altitude to another vertice. You have proven 1/3 of the formula.

Example1: Solve the triangle that has α = 40 degrees, β = 60 degrees, γ = 80 degrees, and side length a is 4 meters meters 60 b 80 c Since we know an angle and its opposite side we can use the Law of Sines.

If you set up the law of sines and get a calculator error, then there is no triangle. If you find the missing angle, and the sum of its supplement and the given angle are less than 180 degrees then there are 2 different triangles. If you find the missing angle, and the sum of its supplement and the given angle are greater than 180 degrees then there is 1 triangle.

Example 1: a = 3 b =2, and alpha = 40 degrees Answer: beta = 25.4 degrees Supplement of beta is degrees The supplement (154.6) and the given angle (40) has a sum that is greater than 180 degrees, Therefore there is one triangle β 2 γ c

Example 2: a = 6 b = 8, and alpha = 35 degrees Answer: beta = 49.9 degrees Supplement of beta is degrees The supplement (130.1) and the given angle (35) has a sum that is less than 180 degrees, Therefore there is two triangles β γ c

Example 2: a = 6 b = 8, and alpha = 35 degrees Since beta could be 49.9 degrees or degrees, we have two possible triangles using the given information. Now you can solve the triangles to obtain the values for c and gamma γ c 35 8 c 6 γ 130.1

2 b α β 50 1 Example 3: a = 2 c = 1, and gamma = 50 degrees Answer: Alpha = Error Since there was a calculator error, there is no triangle.

WARM UP Using the Law of Sines Find: Height y

Packet: Page #9,13,21,25,29,31,37-44 all,47,51,57