Warm up sin cos −1 2 2 cos tan −1 − 3 tan −1 tan 2𝜋 3 tan sin −1 3 2 cos −1 sin 7𝜋 6

The Law of Sines and The Law of Cosines

Not to be bleak… We know how to solve a right triangle. But what do we do when the triangle is oblique? You apply the Law of Sines. You can solve an oblique triangle if you know the measures of two angles and a non-included side (AAS), two angles and the included side (ASA) or two sides and a non-included angle (SSA).

Apply the Law of Sines (AAS) Solve ∆ABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Now you try… Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Real world example An Earth-orbiting satellite is passing between the Oak Ridge Laboratory in Tennessee and the Langley Research Center in Virginia, which are 446 miles apart. If the angles of elevation to the satellite from the Oak Ridge and Langley facilities are 58° and 72°, respectively, how far is the satellite from each station?

Real world example Two ships are 250 feet apart and traveling to the same port as shown. Find the distance from the port to each ship.

Problems!!!! …… SSA From geometry, we know that the measures of two sides and a nonincluded angle (SSA) do not necessarily define a unique triangle. Consider the angle and side measures given.

The ambiguous case: acute Consider a triangle in which a, b, and A are given. For the acute case: sin 𝐴= ℎ 𝑏 What does h=? One solution No solution two solution One solution

The ambiguous case: obtuse One solution no solution

How does it work ? Ambiguous case: 0 or 1 solutions Find all solutions for the given triangle, if possible. If not solution exists, write no solution. Round side lengths to nearest tenth and angle measures to nearest degree. 𝑎=15, 𝑐=12, 𝐴=94° Is the angle acute or obtuse? Is a greater than or less than c? Therefore: how many solutions are there? Apply the Law of Sines Plan B: Jump straight to Law of Sines when obtuse and SSA. If it works, great. If not, then no solution.

Ambiguous case: 0 or 1 solutions Find all solutions for the given triangle, if possible. If not solution exists, write no solution. Round side lengths to nearest tenth and angle measures to nearest degree. 𝑎=9, 𝑏=11, 𝐴=61° Is the angle acute or obtuse? Is a greater than or less than b? What is h? Therefore: how many solutions are there? Plan B: The side across from the angle (sharing the letter) is shorter. Jump straight to the Law of Sines. If it works, great. If not, then no solution.

Now you try 𝑎=12 , 𝑏=8 , 𝐵=61° 𝑎=13 , 𝑐=26, 𝐴=30°

Ambiguous case: 2 solutions Find two triangles for which 𝐴=43°, 𝑎=25, and 𝑏=28. Round side lengths to the nearest tenth and angle measures to the nearest degree. Is the angle acute or obtuse? Is a greater than or less than b? What is h? Therefore: how many solutions are there? Angle B will be acute, and angle B’ will be obtuse. Make a reasonable sketch of each triangle and apply the Law of Sines to find each solution.

Solution 1: ∠𝐵 is acute Find B Find C Tell-tell sign that there is a second solution to the SSA case: <B is acute. Which means, again, you can jump straight to the Law of Sines.

Solution 1: ∠𝐵’ is obtuse Find B Find C

Now you try 𝐴=38°, 𝑎=8, 𝑏=10 𝐴=65°, 𝑎=55, 𝑏=57

Law of Cosines You can use the Law of Cosines to solve an oblique triangle for the remaining two cases: when you are given the measures of three sides (SSS) or the measures of two sides and their included angle (SAS).

Real World Example When a hockey player attempts a shot, he is 20ft from the left post of the goal and 24 feet from the right post, as shown. If a regulation hockey goal is 6 feet wide, what is the player’s shot angle to the nearest degree?

Real world example A group of friends who are on a camping trip decide to go on a hike. According to the map shown, what is the angle that is formed by the two trails that lead to the camp?

Apply the Law of Cosines Solve ∆𝐴𝐵𝐶. Round side lengths to the nearest tenth and angle measures to the nearest degree. Use the Law of Cosines to find the missing side measure. Use the Law of Sines to find missing angle measure Find the measure of the remaining angle. b=8 , a=5, angle C= 65ᵒ

Now you try! Solve ∆𝐻𝐽𝐾 if 𝐻=34°, 𝑗=7, 𝑘=10

Area of a triangle given SAS

Find the area given SAS Find the area of ∆𝐺𝐻𝐽 to the nearest tenth.

Find the area given SAS

In a Nutshell If you have a right triangle, stick with SOHCAHTOA and the Pythagorean Theorem If you have AAS, ASA, or SSA, use the Law of Sines If you have SAS or SSS, use the Law of Cosines

Initial Law of Sines: p298 #2-8 Even Ambiguous Case (0 or 1 Soln): p298 #10-18 Even Ambiguous Case (2 Solutions): p298 #20-26 Even Law of Cosines: p298 #28-36 Even