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Day 7: Apply Law of Sines.
http://www.classzone.com/eservices/home/pdf/student/LA213EAD.pdf
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort, student’s unit circles
Time: 50 minutes
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
Objectives
1. Students will be able to determine whether a triangle that is not right can be solved given
AAS, ASA, or SSA.
2. Students will apply the Law of Sines to solve for a triangle.
Learning Activities
1. Attention Grabber: What kind of triangles have we been able to solve for? (right
triangles).
 We can solve a triangle without a right angle as long as we know three parts: Two
angles and any side (AAS or ASA); Two sides and an angle opposite one of them
(SSA); Three sides (SSS);Two sides and their included angle (SAS)
2. We will be looking at the first two cases today.
3. Ask the class to think about what they think it means to have a triangle with AAS, ASA,
or SSA. Have the students write this down on a separate sheet of paper, they should be
encouraged to draw pictures. Students should collaborate after about a minute, will be
turned in at the end of class for participation points.
4. Law of Sines. Have the students draw on the board some triangles that have this
scenarios.
5. Ask the class to solve the following triangle.
6. Students should solve triangle ABC with C = 103°, B = 28°, and b = 26 feet with
partners. Students are should use white boards within pairs to solve together. Be sure that
they take pictures to have in notes. Answers: a ≈ 41.8 feet, c ≈ 54.0 feet, A= 49˚
7. Activity. Solve Triangle ABC with a = 4 inches, b = 2.5 inches, and B = 58°, but students
should get a piece of paper to draw this accurately, and this should be done with pairs.
 Students individually should write a reflection on the back of the paper and
explain what they noticed. This will be turned in along with work done as pairs.
8. Extension: A word problem.
Assessment
Formal: The little project, and text book assignment (will not be due next day but day 9). P. 803
# 21-31 odd, 37, 38, 47, 48, 56
Informal: Participation that is collected on the Excel sheet.
Reflection
Day 8: Apply Law of Sines.
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort, student’s unit circles
Time: 50 minutes
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
Objectives
1. Students will be able to apply the Law of Sines in real world examples.
2. Students will be able to find the area of a triangle that is not right using the Law of Sines.
Learning Activities
1. Attention Grabber: The class writes on the board the problems they have questions on,
from their homework. Students that know how to solve them go to the board and explain
their process to the class. Teacher gives input when needed.
2. Word problem:
 At certain times during the year, you can see Venus in the morning sky. The
distance between Venus and the sun is approximately 67 million miles. The
distance between Earth and the sun is approximately 93 million miles. Estimate
the distance between Venus and Earth if the observed angle between the sun and
Venus is 34°.
 Students should first work independently and then collaborate. When it looks like
most are getting it, go back to a class discussion.
3. Finding the area of a triangle that is not right, using the Law of Sines.
 The area of any triangle is given by one half the
product of the lengths of two sides times the sine
of their included angle. For triangle ABC shown,
there are three ways to calculate the area:
Area = 1/2 bc sin A; Area = 1/2 ac sin B
Area = 1 2 ab sin C
4. Have the class find the area of the following triangle. First let the students look at the
problem independently and then finish it as a class.
5. Extension: For those ready to move on, otherwise the rest of
class will go over questions. You are buying the triangular piece
of land shown. The price of the land is $2000 per acre (1 acre = 4840 square yards). How
much does the land cost?
6. Introduction of the real world assignment due on Review day. They can either find an
article that has trigonometry in it and write a summary, take a picture of a triangle that is
found in the real world and label the sides and angles, they can explain their own word
problem through a Jing video, or anything else that must be okayed by me. It will be
graded as being worth two homework assignments. If there is thought put into it they will
get full credit, if little thought is put into but they did do it they will get half credit.
Assessment
Formal: Textbook assignment and project.
Informal: Participation that is collected on the Excel sheet. Ticket to leave of their extension
either a question they had answered or the word problem answered.
Reflection
Resources:
http://www.classzone.com/eservices/home/pdf/student/LA213EAD.pdf
Day 9: Apply Law of Cosines. Mini project.
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort
Time: 50 minutes
Standards: North Dakota’s
Cluster: Apply trigonometry to general triangles.
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
Objectives:
1. Students will be able to apply the Law of Cosines to find unknown measurement in
triangles.
2. Students will be able to determine when it is possible to use the Law of Cosines given
certain measurements.
3. Students will be able to find the area of a triangle using Heron’s Area Formula.
Learning Activities:
1. Attention grabber: On the board have the question “Now that we know how to solve
measurements in a right triangle, do you think it would be possible to solve unknown
measurements in a non-right triangle?” (comment it would be nice)
2. Let’s say for instance we want to find the distance the Earth has travelled around the
sun after a minute has passed.
-We know the distance the Earth is from the sun is approximately 149,500,000 km
and we also know that in that minute the angle created by the two positions of the
Earth is approximately .0006844626 degrees.
-In this case we are given a side, an angle and a side. So far we do not have any
formula that would tell us the distance the Earth has traveled.
3. Then someone came along and discovered the Law of Cosines, which comes from the
Pythagorean Theorem.
4. Then have on the board the Law of Cosines
 a2 = b2 + c2 – 2bc cosA
 b2 = a2 + c2 – 2ac cosB
 c2 = a2 + b2 – 2ab cosC
5. Have on the board a triangle in SAS case (side, angle, side). A triangle with given
lengths of c = 20 and b = 16. You are also given the angle measure of A to be 44
degrees. Find the length of a.
6. Ask the students which equation we should use, from step 4 (they should choose the
one with the angle measure A).
7. Then have the students walk you through the steps into having to solve for a (the
answer is approximately 13.99).
8. Then put an example on the board of a SSS example. A triangle given the lengths a =
6, b = 9, and c = 7. Have the students find all the angles with the help of one other
classmate, this can be done at their desks or on the white boards. (When figuring out
the last angle have them use the equations still to make sure the angles add up to 180
degrees).
9. Make sure students have this example written down in their notes correctly by going
over someone’s work on the board. (answer: A = 41.75, B = 87.27, and C = 50.98,
41.75 + 87.27 + 50.98 = 180).
10. Go back to the Earth problem and the distance the Earth traveled after one minute.
11. Ask the class what kind of case it is SAS or SSS. (answer (SAS)
12. Have the class work together again on this problem.
 What is the a, b, and c values? (This will differ depending on where they place
their them on the triangle)
A
 Which one are we looking for?
 What angles correspond to each letter A, B, and C?
c
b
 What equation did they choose to use?
 What is the distance? (1785.78 km)
 Could you find the other two angles if necessary?
B
a
13. What is the area of a right triangle? (they should be able to
tell you that it is ½ bh)
 Do you know of a way to find the area of a triangle that is not right?
14. A Greek mathematician Heron discovered a formula which is
√𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where s = ½(a + b + c)
 s is the semiperimeter or half-perimeter of the triangle
15. As a class go through the process of finding the area of a triangle with given side
lengths of a = 13m, b = 4m, and c = 11m. First find s then plug in all the values.
(A=20.49 m2)
16. Extension: Have on the board an example of where they would need to find the area
of a triangle in the real world.
 The intersection of three streets forms a piece of land call a traffic triangle.
The distance from A to B is 100 yards, from B to C is 178 yards, and the
distance from C to A is 293 yards.
 Students will work on this with their partner either in their notebooks or on the
whiteboards.
 First find s, then the area.
Assessment
Formal: Homework assignment from book, and to find or make up a triangle that you
need to find all the sides and angles when given either just SAS or SSS.
Informal: Participation with working with partners and discussing as a class collected on
an Excel sheet.
Reflection
Algebra II 9th grade
The lesson I taught today is for two classes. In this class I could have spent more time
explaining as a whole class. I gave a lot of time for the students to come up with how to solve the
problems, but more examples as a class would have made that time more beneficial. Also, if I
had a better feel of the technology the lesson would have gone a lot smoother. I did not feel I was
very enthusiastic either which would have helped the class look more engaged. A thing I felt was
C
good was that students were not afraid to tell me I was doing something wrong. In the class I did
not get through everything. Students came in after class for help.
11th grade
I felt this lesson went a lot better. I explained the material better and clearer, I got through
everything with extra time for students to start their homework. One thing I could have done
better was stop for questions and ask about S.S.S. situations and S.A.S. situations. The
collaboration in this class was not as good as the previous class, but everyone had the same issue
when solving the real world problem, so I had a chance to explain it to the whole class. Overall
the class was very quiet, and they would say the answer to a question very quietly.
References:
http://www.overthinkingit.com/wp-content/uploads/2009/01/figure_3.jpg
Name____________________
Directions: Find a triangle or draw your own triangle (examples: the legs of a chair, a roof of a
dog house, folded paper, etc.). Measure the sides of the triangle so you have all 3 side
measurements that are labeled. Find all three angles A, B, and C using the Law of Cosines. To get
full credit describe the object you found (draw a picture of it), provide accurate measurements (rounded
to 2 decimal places), and show all of your work. If you are creating your own triangle, measure the exact
length. If a ruler is not present, be sure to state what you are using as your form of measurement.
Day 10: Quiz on law of sines and law of cosines.
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort, students should have a
reading book
Time: 50 minutes
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
Objectives
1. Students will be able to determine which triangles can be solved for.
2. Students will be able to determine when to use Law of Sines or when to use Law of
Cosines given sides and angles of a triangle.
3. Students will be able to solve a triangle using either Law of Sines or Law of Cosines.
Learning Activities
1. Attention Grabber: The class writes on the board the problems they have questions on,
from their homework. Students that know how to solve them go to the board and explain
their process to the class. Teacher gives input when needed.
2. Finish up lesson the Law of Cosines.
3. Any questions?
4. Pass out quiz.
5. Correct the quiz in class, students will correct their own, but a different writing utensil
will be used (I will hand out markers).
 Problems should be gone over as a class and if students have any questions they
can be answered.
6. Extension: Unit circles should be completed, and graded (checked off that they were
completed).
Assessment
Formal: Quiz
Informal: Participation and unit circles observed and collected in the Excel sheet for
participation.
Reflection
Name ___________________
Algebra II - Trigonometry Quiz 3
1. Which law would you use given SAS?
Law of Cosines
2. Which law would you use given SSS?
Law of Cosines
3. Are there any combinations of sides and angles that would not allow you to solve the
triangle? If so name a combination.
Yes, when only given ASS
4. Solve the following triangles.
(a)
a ≈ 5.3; B ≈ 45.41; C ≈ 85.59
(a)
B=95; b = 15.94; a = 13.12
Day 11: A mix review on everything from the unit.
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort
Time: 50 minutes
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
HS.FTF.6
(+) Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
HS.FTF.7*
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context
HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.*
HS.FTF.1
Understand that the radian measure of an angle is the length of the arc on the unit circle
subtended by the angle.
HS.FTF.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
Objectives
Learning Activities
1. Attention Grabber: Open the floor to any question about the chapter.
2. Split the class into groups divided into groups based on what students had questions on.
3. Groups should be focused on different parts of the material, and the teacher should be
going around and answering any questions there might be.
4. Extension: Math video on TedEd.
Assessment
Informal: participation graded on Excel sheet.
Reflection
Day 12: Test over the unit.
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort, a book for reading
Time: 50 minutes
Standards
HS.GSRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
HS.FTF.6
(+) Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
HS.FTF.7*
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context
HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.*
HS.FTF.1
Understand that the radian measure of an angle is the length of the arc on the unit circle
subtended by the angle.
HS.FTF.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
Objectives
Grade: 9
Content: Mathematics-Algebra II
Materials: pencil, paper, whiteboard and markers, projector of some sort
Time: 50 minutes
Learning Activities
1. Hand out the test.
2. When students get done they should have a reading book with to read, or they will be
given a math article to read and summarize.
Assessment
Formal: Chapter Test
Reflection
Answer Key
Trigonometric Ratios and Functions Unit Test
Write out True or False for the following. If it is false correct it to make it a true statement.
1. The hypotenuse is the longest side of a right triangle. True
2. An angle is in standard position if its vertex is at the origin and its initial side lies on the
positive x-axis. True
3. The value of sin 𝜃 corresponds to the x value on the unit circle. False y
1
𝜋
𝜋
4. The inverse of 𝑐𝑜𝑠 −1 2 is 4 . False 3
5. The law of sines can be used to solve triangles when two angles and the length of any
sides are known. True
6. You cannot solve a triangle when only given the lengths of two sides and angle opposite
one of the two sides are known. False you can using the law of sines
7. The following equation 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴 can be used when given three known
lengths of a triangle. True
Short answers. Show all work in space provided.
8. 𝑆𝑖𝑛𝜃 is defined as
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
9. 𝐶𝑜𝑠𝜃 is defined as
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
4
10. In a right triangle, 𝜃 is an acute angle and cos = 9. What is the value of tan 𝜃? Show all
your work to receive full credit. (leave as a fraction
√65
4
11. Which angle measure is shown in the diagram?
140
12. You are given the point (2,-2) on the terminal side of an angle 𝜃in standard position,
evaluate cosine. To be given full credit, draw a picture. Picture should look similar to 11
√2
2
13. Evaluate 𝑡𝑎𝑛−1 2.6 in both radians and degrees. Round to two decimal places.
1.20; 68.96
14. Which triangles can be solved using Law of Sines?
Angle Angle Side
Angle Side Angle
Side Side Angle
Match the following with their correct responses in the right column. Use the following triangle
when necessary.
180°
_____15. Which case
h
a.
_____16. Angle B
c
b. 45.94
_____17. Angle C
g
c. 27.23
_____18. The missing length
d
d. 28.88
_____19. The semiperimeter
b
e. 277.51
_____20. The area
e
f.
_____21. Define Tan 𝜃
j
g. 112.12
_____22. Define Cot 𝜃
f
h. SAS
_____23. Degree to radians
i
i.
_____24. Radians to degrees
a
j.
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
180°
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
25. Convert -260 to radians.
−
13𝜋
9
26. Write tan 𝜃 as the ratio of two other trigonometric functions..
sin 𝜃
cos 𝜃
27. You are building a triangular concrete patio that has sides of length 8 feet, 11 feet, and 15
feet, and a thickness of .5 foot. If one bag of cement makes .33 cubic foot of concrete, how many
bags of cement do you need to build the patio? What are you going to use to solve this? Explain
your reasoning to your answer in 4 or more complete sentences.
The answer is 65 bags of cement. This is because you need 64.92 and so you need to round up so
there is enough cement. I went about this problem using Heron’s Area formula. I knew I had to
use something that did not use right triangles since there was no mention of a right triangle. Once
I found the surface area I multiplied it by the thickness of the patio, and then divided it by .33 ft3,
the amount in each bag.
28. What is one thing that you liked about this unit? What is one thing that you still do not
understand? Answer should be 2-3 sentences.
Answers will vary