Law of Sines and Cosines

Oblique TriangleS Oblique triangle—a triangle that does not include a right angle. Laws of Sines and Cosines used to find missing parts of oblique triangles.

Law of Sines Used for ASA AAS SAA SSA—ambiguous case

Law of Sines 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶

Solving a SAA Triangle Given 𝐴=46°, 𝐶=63°, 𝑎𝑛𝑑 𝑐=56 𝑖𝑛. Solve the triangle rounding lengths of sides to the nearest tenth.

Another One. . . Given 𝐴=64°, 𝐶=82°, 𝑎𝑛𝑑 𝑐=14. Solve the triangle.

Solving an ASA TrianglE Given 𝐴=50°, 𝐶=33.5°, 𝑎𝑛𝑑 𝑏=76, solve the triangle rounding all measures to the nearest tenth.

Another one of Those. . . Given 𝐴=40°, 𝐶=22.5°, 𝑎𝑛𝑑 𝑏=12.

Law of Cosines Used for SSS and SAS—angle included (Remember SSA is ambiguous case for Law of Sines.)

The Law In Words: The square of a side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of their included angle. (Whew!!!)

The Law 𝑏 2 = 𝑎 2 + 𝑐 2 −2𝑎𝑐 cos 𝐵 𝑐 2 = 𝑎 2 + 𝑏 2 −2𝑎𝑏 cos 𝐶 In formula: 𝑎 2 = 𝑏 2 + 𝑐 2 −2𝑏𝑐 cos 𝐴 𝑏 2 = 𝑎 2 + 𝑐 2 −2𝑎𝑐 cos 𝐵 𝑐 2 = 𝑎 2 + 𝑏 2 −2𝑎𝑏 cos 𝐶 (much better!!)

Solving an SAS Triangle Use the Law of Cosines to find the side opposite the given angle. Use the Law of Sines to find the angle opposite the shorter of the two given sides (always acute). Find the third angle by subtraction.

SAS Example 𝐴=60°, 𝑏=20, 𝑎𝑛𝑑 𝑐=30

Another SAS 𝐴=120°, 𝑏=7, 𝑎𝑛𝑑 𝑐=8.

Solving an SSS Triangle Use the law of Cosines to find the angle opposite the longest side. Use the Law of Sines to find either of the two remaining acute angles. Find the third angle by subtraction. BEWARE OF SIDES THAT ARE TOO SHORT. REMEMBER THE SUM OF ANY TWO SIDES MUST BE LONGER THAN THE THIRD SIDE OR NO TRIANGLE EXISTS!!

SSS Example 𝑎=6, 𝑏=9, 𝑎𝑛𝑑 𝑐=4

Another SSS 𝑎=8, 𝑏=10, 𝑎𝑛𝑑 𝑐=5

Applications of laws of sines and cosines 1. The Leaning Tower of Pisa in Italy leans at an angle of about 84.7 degrees. At a point 171 feet from the base of the tower, the angle of elevation to the top is 50 degrees. Find the distance, to the nearest tenth of a foot, from the base to the top of the tower.

2. A surveyor needs to determine the distance between two points that lie on opposite banks of a river. He measures 300 yards from the first point on one bank to another point on the same bank. The angles from each of these points to the point on the opposite bank are 62 degrees and 53 degrees. Find the distance between the original point and the point on the opposite bank.

3. A Major League baseball diamond has four bases forming a square whose sides measure 90 feet each. The pitcher's mound is 60.5 feet from home plate on a line joining home plate and second base. Find the distance from the pitcher's mound to first base.

4. An architect's client wants to build a home based on the architect Joh Lautner's Sheats-Goldstein House--a house by a famous architect. The length of the patio will be 60 feet. The left side of the roof will be at a 49 degree angle of elevation, and the right side will be at an 18 degree angle of elevation. Determine the lengths of the left and right sides of the roof and the angle at which they will meet.

5. Lola rolls a ball on the ground at an angle of 23 degrees to the right of her dog Buttons. If the ball rolls a total of 48 feet, and she is standing 30 feet away, how far will Buttons have to run to retrieve the ball?

Finding area using trig

Example for sas case

Draw it first! Find the area of an equilateral triangle with one side length 6.

Using the sss case

Draw it A piece of commercial real estate is priced at $3.50 per square foot. Find the cost, to the nearest dollar, of a triangular lot measuring 240 feet by 300 feet by 420 feet.