Applications of Trigonometry and Matrices Unit 8

Law of Sines Lesson 8.1

What if it is not Right? All of the trigonometry problems discussed in the previous unit deal with Right Triangles. But how do we solve for missing pieces in a triangle that isn’t a Right Triangle??? THE LAW OF SINES is one way!

Law of Sines

Examples: In the triangle ABC, given A = 32°, B = 72°, and a = 12.5, find the missing pieces of the triangle. In triangle ABC, given A = 48°, C = 59°, and b = 24, find the missing pieces of the triangle.

Example: In triangle ABC, given A = 38.5°, a = 18.3, and b = 25.2, find the missing pieces for the triangle.

Example: In triangle ABC, given A = 68.5°, a = 105, and c = 135, find the missing pieces for the triangle.

What happens with Law of Sines, stays with Law of Sines. When using the Law of Sines, things can get crazy! 1) When given two angles and a side that corresponds to one of the angles, you will always have one triangle. 2) When given two sides and one of the corresponding angles, it is possible to have two triangles (Ambiguous Case). 3) When solving for a missing angle, the inverse sine cannot be taken of a number greater than 1 or less than negative one, which means the triangle cannot exist.

Homework: Pages 569 – 571 #’s 1, 5, 7, 11, 15, 17, 19, 21, 23, 27

Bell Work: 1) In triangle ABC, given A = 42.5°, a = 16.3, and b = 21.7, find the missing pieces of the triangle. 2) Billy and Anna are standing on opposite end zones of a football field (100 yards apart). A kite is being flown between them. The angle of elevation from Anna to the kite is 34°, and from Billy to the kite is 41°. How far off the ground is the kite?

Word Problem: A plane has a heading of N35°W and is traveling 320 mph. After traveling for two hours, the plane shifts its heading to S21°W and continues to travel for another hour and a half. What is distance from the planes starting point to its end point?

Law of Cosines Lesson 8.2

Why do we need another LAW? If we do not have a side and a corresponding angle, then we are unable to use the Law of Sines. The Law of Cosines can be used when: A) We know all three sides of the triangle. B) We know two sides and a third angle that corresponds to the missing side.

Law of Cosines (3 for 1 special!!!)

Examples: 1) Given a = 7.1, b = 8.4, and C = 71.4°, solve for the missing pieces of triangle ABC. 2) Given a = 33, b = 44, and c = 55, solve for the missing pieces of triangle ABC.

Rearranging the Law of Cosines:

Homework: Pages #’s 1 – 23 odds

Bell Work: Given a = 14.5, b = 11.2, and c = 17.1, solve for the missing pieces of triangle ABC.

Example: Kyle and Spencer run 7.5 miles together heading south. After the seven mile run, they then run another 5.3 miles heading N32°W. How far away are they from their original starting point?

Example: Kearstyn runs 2.1 miles N30°E, then runs 4.2 S Ɵ E where Ɵ is an acute angle. She estimates that she is now 5.3 miles from her starting point. What direction should she run in to reach her starting point?

Homework: Try these if you need: Pages 579 – 580 #’s 12 – 24 evens

Example: A parallelogram has sides of 25 cm and 50 cm, with one interior angle measuring 54°. What is the approximate length of the diagonals?

Area of a Triangle What is the area of a triangle? Find the area of triangle ABC if you know that a = 22 cm, b = 27 cm, and C = 41°.

Area of a Triangle:

Examples: Find the area of triangle ABC if you know a = 45m, b = 32m, and A = 61°. Find the area of triangle ABC if you know a = 12.8ft, b = 15.3ft, and c = 17.9ft.

Homework: Pages 580 – 581 #’s 24, 29 – 37 odds

Bell Work: 1) If a = 20, b = 30, and c = 40, solve for the missing pieces of triangle ABC. 2) What is the area of triangle ABC from number 1?

Word Problem: A boat leaves port at 1:00 P.M. and heads N25°W traveling 12 mph. A second boat leaves port at 2:00 P.M. traveling 16 mph heading S25°W. How far apart are the boats at 3:30 P.M.?

Word Problem: A helicopter travels 120 miles from point A in the direction 115°. The helicopter then travels 90 miles from point B in the direction of 220°. How far will the helicopter be from point A?

Review for Quiz Law of Sines Law of Cosines Area of Triangles Pages 620 – 623 #’s 5 – 10 (approximate), 40, 42, 43, 47

Bell Work:

Matrix Algebra And yes, this is just like the movies… Bonus Lesson!!!

What is a Matrix??? A matrix is a collection of numbers or elements put into order of rows and columns (m x n). We can perform different operations with matrices, and they are used in upper level math to help organize complex computations of polynomials or equations.

Addition and Subtraction of Matrices: We can add and subtract matrices by simply adding and subtracting the corresponding elements in the matrices. In order to add or subtract, the matrices must be the same size! (must have the same number of rows and columns)

Matrix Properties If A, B, and C are m x n matrices: 1) A + B = B + A (they are commutative) 2) A + O = A (O is the zero matrix) 3) A + (-A) = O (matrices have an additive inverse) 4) cA, where c is a real number, would result in multiplying every element of A by the number c.

Matrix Multiplication!!! This is awesome… 1 st !!! In order to multiply two matrices together, the columns in the first matrix must match the rows in the second matrix. The number of rows in the first matrix and the number of columns in the second matrix tells us the size of the resulting matrix. We can multiply a 2x3 matrix by a 3x5 matrix. Result: 2x5 We can multiply a 4x2 matrix by a 2x7 matrix. Result: 4x7

Matrix Multiplication: To multiply two matrices, A and B, multiply each row in matrix A by each column in matrix B. The explanation of how to multiply matrices is very confusing to write down, it is much easier to understand by following examples and practicing. Just remember: multiply a row by a column, and find the sum of all of the products, and that row/column tells which element to fill in! Sounds easy.

Homework: Pages 685 – 686 #’s odds

Bell Work: Problems are on the chalkboard!

Matrix Multiplication Continued: For two matrices, A and B, would the following statements be true or false? (assuming they are able to be multiplied) AB = BA (2A)B = 2(AB) (A + B)(A – B) = A² – B²

Classwork/Homework: Pages 685 – 687 #’s 12 – 26, 35, 37, 39 Definitely try #’s 35, 37, and 39!!!

Bell Work: Use the two matrices on the board to find the following: 1) 4A – 2B 2) AB 3) BA 4) Would A(B + C) = AB + AC ??????

Class Work/ Home Work! Page 622 #’s 44, 45, 48

Bell Work: Mr. Kelsey is on a plane that has been taken over by Math Terrorists. The planes original heading was supposed to be S43°E with a destination that was 300 miles away. The Math Terrorists took over the plane and managed to travel 110 miles with a heading of S70°E before Mr. Kelsey regained control of the plane. Since the pilots are incapacitated, Mr. Kelsey has to fly the plane to its original destination. How far is the plane away from its original destination and what heading should Mr. Kelsey set for the plane?

Example: At 12:00 P.M., Colten rides his snowmobile 30 mph with a heading of S80°W. 30 minutes later, Jimmy leaves from the same spot on his snowmobile, heading N10°E traveling 35 mph. How far apart are Colten and Jimmy at 3:00 P.M.?

Example: Kylan decides to go for a run one morning. He runs 6 mph with a heading of N40°E for 45 minutes, then changes direction (keeping the same speed) with a heading of S32°E. If he spent a total of 2 hours running, how far away will Kylan be from his original starting point.

Small Test Tomorrow Law of Sines Law of Cosines Matrices (Addition/Subtraction/Multiplication)