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Unit 8
Applications of Trigonometry and Matrices
Lesson 8.1
Law of Sines
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
𝑠𝑖𝑛𝐴

𝑎
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
𝑐
 Where A, B, and C are angles in the triangle and a, b, and c
are the sides that correspond to the angles.
 *Please note that the book uses 𝛼, 𝛽, 𝑎𝑛𝑑 𝛾 as the angles.
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?
Lesson 8.2
Law of Cosines
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!!!)
 In triangle ABC,
 1) 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 ∙ 𝑐𝑜𝑠𝐴
 1) 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 ∙ 𝑐𝑜𝑠𝐵
 1) 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 ∙ 𝑐𝑜𝑠𝐶
 Where a, b, and c are the sides of the triangle and A, B, and C
are the corresponding angles.
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:
 When you are using the Law of Cosines to solve for a missing
angle, rewrite the original formula so that you just have to
substitute the values in.
 Ex: 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 ∙ 𝑐𝑜𝑠𝐴 can be rewritten as:
 𝑐𝑜𝑠𝐴 =
𝑎2 −𝑏2 −𝑐 2
, then
−2𝑏𝑐
just take the inverse cosine.
Homework:
 Pages 578-580 #’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 SƟE
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:
 Two ways:
 1) Area of a triangle equals one half the product of the
lengths of any two sides and the sine of the angle between
them.
 2) Heron’s Formula: Area =
 Where 𝑠 =
1
(𝑎
2
𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
+ 𝑏 + 𝑐). (𝑠 = half the perimeter)
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:
 DO NOT SOLVE! Just think about how you would solve
this. Think back to Algebra II.
 5𝑥 − 4𝑦 − 3𝑧 = 10
 3𝑥 − 𝑦 + 4𝑧 = 20
 𝑥 + 4𝑦 − 4𝑧 = 30
Bonus Lesson!!!
Matrix Algebra
And yes, this is just like the movies…
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…
 1st!!! 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 1- 25 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)