Youngstown City Schools
MATH: GEOMETRY
UNIT 2B: TRIGONOMETRY (6 WEEKS) 2013-2014
Synopsis: Students will study and expand upon the concept of trigonometry, starting with the connection to similar triangles and ending with real life applications using the law of sines and cosines. Due to the length of this unit, it will be broken into three sections with an assessment after each section.STANDARDS
G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*
G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*
G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*
G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.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).
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
L.2 Communicate using correct mathematical terminology
L.5 Justify orally and in writing mathematical reasoning
MOTIVATION / TEACHER NOTES1. Show the you tube video on building clinometers
http://www.youtube.com/watch?v=CsNbfxDQnYM. Explain to students that they will be building their own clinometers and using them to calculate heights of objects later in the unit. (G.SRT.8, MP.2, MP.4, MP.5, MP.7, L.2)
2. Discuss uses of trigonometry in real life: surveyors and civil engineers use it to calculate distances and angles; carpenters use it to find missing lengths when building a house; electrical engineers who test speakers use it to achieve maximum performance from their speakers; manufacturing products technologies use it to determine the important angles of manufacturing tools; it is used in geography and navigation by sailors to determine their positions when they were in the middle of the sea; astronomers use it to calculate the position of the planets; and the geographical concept of latitude and longitude are also applications of trigonometry. (G.SRT.8, MP.4, L.2)
3. Preview expectations for end of Unit
4. Have students set both personal and academic goals for this Unit or grading period.
TEACHING-LEARNING / TEACHER NOTES /
Vocabulary
Sine / Trigonometry / Opposite side / Leg
Cosine / Radicals / Adjacent side / Complementary angle
Clinometers / Altitude / Hypotenuse / Tangent
1. Start the introduction to trigonometry with the activity in the textbook on page 365, then extend it to include angle C also. After this is completed, explain to the students the decimal representation for cos 22° is calculated by taking a right triangle with a 22° angle, measuring the side adjacent to the angle and the hypotenuse, creating a ratio of , and divide to get the decimal approximation. Every time you take a right triangle with a 22° angle and set up this ratio, it will always give the same result regardless of the length of the sides because the triangles are similar. This is a great deal of work, so mathematicians made a trig table to make calculations with angles and sides of triangles easier. Show students trig table on the web site: http://www.classzone.com/cz/books/pre_alg/resources/pdfs/formulas_and_tables/palg_table_of_trig_ratios.pdf. Now, of course, this is programmed into calculators and trig tables are no longer needed. (G.SRT.6, MP.2, MP.4, MP.8, L.2)
2. Review the terminology using pictures of right triangles: trigonometry, side (leg) adjacent, side (leg) opposite, hypotenuse, leg, complementary angles. (G.SRT.6, MP.4, L.2)
3. Student activity: draw right triangle ABC with AC perpendicular to AB. Have students fill in the table and explain the relationship between angle C and angle B: (G.SRT.6, MP.2, MP.4, MP.7, L.2, L.5)
Angle / Side opposite / Side adjacent / Hypotenuse
B
C
4. Activity: Solve the following using similar triangles, in ∆ABC, AC is perpendicular to BC and in ∆RST,RS is perpendicular to ST.
1. ∆ABC∆RST, BC = 3, AB = 5, RS = 15, find TS (ans. 9)
2. ∆ABC∆RST, AC = 9, AB = 14, RS = 4, find RT (ans. 2.6)
After students solve these problems, state the definition of sine, cosine and tangent ratios. Then, ask them if there is an easier method of solving the above problems. Use the trig tables to show them for the first problem the left hand side (ratio) is the same as cos 53° and the problem can be solved very simply by multiplying cos 53° by 15 to find TS. Likewise, the left hand side (ratio) of the second problem is the same as sin 40° (looking at the trig tables), so this problem can be solved by multiplying 40° by 4 to find RT. (G.SRT.6, MP.2, MP.3, MP.4, MP.6, MP,8, L.2, L.5)
5. Show students how to use the calculator when working with trig functions. Make sure the calculator is in the degree mode before beginning. Compare these values to the values on the trig tables and discuss similarities and differences. Have students fill in the chart below:
Angle / Cosine / Sine
50
40
30
60
70
20
25
65
Discuss the pairs of angles and their relationship to each other. Then question students about the values of the cosine and sine. They should reach the conjecture that the cosine of an angle is equal to the sine of its complement and the sine of an angle is equal to the cosine of the complement. Discuss the reasoning for this with diagrams of right triangles. (G.SRT.6, G.SRT.7, MP.2, MP.3, MP.4, MP.8, L.2, L.5)
6. Reinforce with the following examples: (G.SRT.6, G.SRT.8, MP.1, MP.4, MP.5, MP.6, MP.8, L.1, L.2)
a) A wire attached to a pole makes a 63° with the ground and is 12′ from the base of the pole. Find the height of the pole. (Ans. 23.5′)
b) A roof truss is in the shape of an isosceles triangle. The base angles are 25° and the equal sides are 10′ each. Find the height of the truss (triangle). (Ans. 4.2′)
c) A road is going up a mountain and makes a 28° angle with the horizontal. How high would you have to rise in going 250 meters up the road? (Ans. 11.7 meters)
d) A 10 foot log is leaning against a barn and makes a 54° angle with the ground. How far is the log from the foot of the barn? (Ans. 5.9 feet)
e) A wire 25 ft. long is supporting an 18 ft. pole. Find the angle formed by the wire and the pole. (Ans. 43.9°)
7. Reinforce with Kuta worksheets (G.SRT.6, G.SRT.8, MP.1, MP.4, MP.5, MP.6, MP.8, L.1, L.2)
1. Trigonometric ratios http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/9-Trigonometric%20Ratios.pdf
2. Solving for sides of right triangles http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/9-Solving%20Right%20Triangles.pdf
3. Solving for angles of right triangles http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Trigonometry%20to%20Find%20Angle%20Measures.pdf
8. To reinforce real-life problems, use section 7-5 in the textbook. Make sure the angles of elevation and depression are discussed which are also found in chapter 7. (G.SRT.8, MP.1, MP.4, MP.5, MP.6, MP.8, L.1, L.2)
9. Before starting the clinometer activity, revisit step 1 of the motivational activity. Create the clinometers using the following web site and the video in the motivational activity:
http://repository-intralibrary.leedsmet.ac.uk/open_virtual_file_path/i1442n87724t/shapes-theod2_clinometer.pdf
Pass out worksheet #1 (attached on page 12) which is the clinometers project. Have students complete the project. (G.SRT.8, MP.1, MP.4, MP.5, MP.6, MP.8, L.1, L.2)
10. Review simplifying radicals. Students should be able to simplify the following radicals before working on 30-60-90 and 45-45-90 triangles: , , , , , ,
11. To begin working on the 45-45-90 triangles (isosceles right triangles) and the relationship between the sides, have students derive the relationship using the Pythagorean theorem and letting the two equal sides be one. If they don’t see the relationship after one example, have them do several more, letting the equal sides be 3, 4, etc. To reinforce, work a few problems using the Kuta worksheet problems 1- 6: (G.SRT.8, MP.4, MP.7, MP.8, L.2) http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8-Special%20Right%20Triangles.pdf
12. Discovery activity with equilateral triangles that have a perpendicular bisector (altitude):
Ask students if they want to work with fractions or whole numbers. Of course they are going to say whole numbers, so call the sides of the equilateral triangle 2x. Then have the students use the Pythagorean theorem to find the length of the perpendicular bisector, leaving the answer in simple radical form. Discuss the angle measures (30-60-90) and the relationship between the sides. To reinforce have students work on the following: (G.SRT.8, MP.1, MP.2, MP.4, MP.6, MP.7, MP.8, L.1, L.2)
a) A piece of tile is in the shape of an isosceles trapezoid having base angles of 60°, and bases 10 and 16. Find the height and legs of the trapezoid. (Ans. Height = 3 and legs are 6 each)
b) A telephone pole is 24 ft. high with a guy wire attached to the top of it. The guy wire makes a 60° angle with the ground. How far is the wire from the base of the pole? (Ans. 8 )
c)
B
(Ans. BC = 6, DB = 6, AD = 4, AB=
2, AC = 8, <BDC = 60, <BDA = 30,
D C <A = 60)
d) http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8- Special%20Right%20Triangles.pdf problems 7-18
13. Discuss example #11 on page 360 in the textbook and then have students create their own problems using 45-45-90 or 30-60-90 relationships. (G.SRT.8, MP.1, MP.2, MP.4, MP.8, L.1, L.2)
Have students take test #1 on standards G.SRT.6, G.SRT.7, and G.SRT.8
14. Geometry is found in everyday life. To assist students with this, find some objects that are represented by geometric shapes and create problems from them. Listed below are two examples of this: (G.MG.1, MP.1, MP.4, MP.5, MP.6, MP.7, MP.8, L.2)
a) Using a rectangular table top with dimensions 10 in. by 20 in. Find the length of the diagonal and the angle between the diagonal and the 20 in. side. (Ans. 10, 26.6°)
b) TV’s are measured by the diagonal. What size TV should you purchase if your cabinet is 40 in. and the angle between the diagonal and the 40 in. side is 29°? (Ans. 45 in. TV)
15. Area is a concept that is also prevalent in our lives today, using it to purchase carpet, paint a room, and seed a yard. Review the area of a triangle (Area = ½ b*h) and later, we will extend it to trigonometry. Start with the following examples: (G.MG.1, MP.1, MP.2, MP.4, MP.5, MP.6, MP.8, L.1, L.2)
a) Find the area of a triangular entrance way that needs to be tiled. The entrance way is a right triangle with hypotenuse 8 ft. and length of one side 6 ft. (Ans. 6 ).
b) A company is building signs in the shape of right triangles as shown below. To prepare to paint them, they need to know the area of the triangles. Find the area.
c.
(Ans. 78.75 sq. in.)
16. Population density is an important concept used by statisticians: Examples of population density problem for students to work on in groups and present solutions to the class are:
a. The YSU stadium has 20,630 seats. The dimension of sections 14-18 is 240 ft. by 160 ft. and contains 3,630 seats. The distance from Petey’s white ball on his hat to the back edge of the sections is 210 ft. Determine how many people a player can see if he is standing on the white ball of Petey’s hat (see drawing below).
(Solution: Area of large triangle: 25200 sq. ft., base of small triangle: 57 ft., area of small triangle: 1425, subtract the two areas to get area of the trapezoid (or use ½ h(b1 + b2 ), use proportion to find the number of people: 2,247 people)
b. Schushsville is a triangular shaped island off the coast of Northville. Two sides of the island are 100 miles and 350 miles with a 24° between them. There are currently 250,000 inhabitants on the island. Last year, there were 12,000 new children born and 10,000 people were recorded as deceased. It is believed that the island could support a population as dense as 150 people/square mile. What is the current population density and what do you expect will happen to the density as time goes on? Hint: to find the area, draw an altitude perpendicular to either given side. (Ans: density is 35.4 people per sq. mi. and if this trend continues, the density should increase gradually).