Advanced Math II
Unit 5: Triangle Trigonometry
Time Frame: Approximately 2.5 weeks
Unit Description
This unit covers all aspects of triangle trigonometry. It begins with a review of the right triangle ratios. The Laws of Sine and Cosine are introduced so that problems involving oblique triangles can be solved. The unit ends with some real-life applications of triangle trigonometry.
Student Understandings
The students will know how to solve triangles using either the right triangle ratios or the
Laws of Sine and Cosine. They will be able to solve real-life problems and to write their answers in degrees, minutes, and seconds or with the decimal equivalent.
Guiding Questions
1.
Can students express angles in degrees, minutes, and seconds and convert such a measurement to a decimal value?
2.
Can students use right triangle trigonometry to find the missing information in a triangle and solve real-life problems involving right triangles?
3.
Can students use the law of sines and law of cosines to solve problems?
4.
Do students recognize when the given information results in a unique triangle and when it does not?
5.
Can the student use trigonometry to find the areas of triangles?
6.
Can students carry out experiments and solve problems using trigonometry?
7.
Can students use vector addition to solve trigonometry problems?
Unit 5 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Geometry
11.
12.
14.
16
Calculate angle measures in degrees, minutes, and seconds (M-1-H)
Explain the unit circle basis for radian measure and show its relationship to degree measure of angles (M-1-H)
Use the Law of Sines and the Law of Cosines to solve problems involving triangle measurements (M-4-H)
Represent translations, reflections, rotations, and dilations of plane figures using sketches, coordinates, vectors, and matrices (G-3-H)
Advanced Math II  Unit 5  Triangle Trigonometry 58

Sample Activities
Activity 1: A Review of Right Triangle Trigonometry (GLE: 11)
This activity begins with a review of right triangle trigonometry studied in Geometry.
Students should be able to
 apply the Pythagorean theorem in both abstract and real-life settings
 define sine, cosine, and tangent in ratio form and calculate them using technology
 model and use trigonometric ratios to solve problems involving right triangles
They should be able to identify an included angle as well as the side opposite and the side adjacent to an angle in a right triangle setting. Students should review the properties of the
30-60-90 degree and 45-45-90 degree triangle. They should know how to find the exact values of the sine, cosine, and tangent ratios for each of those angles. This is also an ideal time to review with students some of the properties of geometric figures that might be used as part of a triangle trigonometry problem.
Give students a handout with various geometric shapes containing right triangles and have them solve for the missing sides or angles. Have them write angles in degrees, minutes, and seconds and sides to the nearest hundredth.
Activity 2: Applications Involving Right Triangles (GLE: 11)
A common use for trigonometry is to measure angles and distances that are either awkward or impossible to measure by ordinary means. Finding heights of buildings, trees, and mountains, finding distances between objects, and finding angles of elevation and depression are a few examples.
For each of the problems, have students
 draw a diagram
 label the known information
 write the trigonometric ratio needed
 solve using the calculator
1. At 2 p.m. a building 450 feet high casts a shadow 75 feet long. What is the angle of elevation of the sun?
2. To measure the height of a building, two sightings were taken 50 feet apart. The first angle of elevation is 45 o
and the second is 30 o
. How tall is the building?
3. The city plans to install a wheelchair ramp at the entrance to its courthouse. The ramp should have a running slope of 1:12 units. The final height of the ramp is 5 feet. What angle of elevation will the ramp have and how long will it be?
4. A lighthouse keeper observes that there is a 3 o
angle of depression between the horizontal and the line of sight to a ship. If the keeper is 58 feet above the water, how far is the ship from shore?
5. The legs of an isosceles triangle are 8 inches long and the angle between them is 35 o
.
What is the length of the third side?
Advanced Math II  Unit 5  Triangle Trigonometry 59

Solutions:
1. 80 o 32’ 16”
2. 68.3 feet
3. the angle of elevation is 4 o 45” 49’ and the ramp will be 60.2 feet.
4. 1106.7 feet
5. 4.8 inches
In the next two activities, have students extend their knowledge to solving oblique triangles.
The first activity (Activity 3) covers those triangles where one side and two angles are known
(the ASA and AAS theorems from Geometry) as well as those triangles where two sides and an angle opposite one of them is given (SSA). SSA information can result in 0, 1, or 2 triangles. The Law of Sines is used in this activity. The second activity (Activity 4) covers those triangles where two sides and an included angle or three sides are known. The Law of
Cosines is used in this activity. Areas of oblique triangles are also covered in activity 4.
Activity 3: Using the Law of Sines (GLEs: 11, 14)
Help students understand that the Law of Sines is used to solve triangles for which two angles and one side are given. Students learn in Geometry that this information always results in a unique triangle. The Law of Sines is also used with what is called the ambiguous case. This applies to triangles for which two sides and the angle opposite one of them are known. It is called the ambiguous case because the given information can result in one triangle, two triangles, or no triangle. The Law of Sines can be used to give the correct conclusion.
Problems:
1. Use the Law of Sines to fill in the following table. Have students a) Sketch each triangle b) Show the setup to be used
Problem # a b c

A

B

C
1. 2 40
o
20
o
2. 2 40
o
40
o
3.
4.
5.
2
4 5 60
o
10
o
100
o
4 6 20
o
2. Have students work #2 from Activity 1 using the Law of Sines and then discuss which method is easier to use.
3. From points A and B, 150 meters apart, a marker is sighted on the opposite side of a river.
From point A the angle made by the line from A to B and the line of sight to the marker is 80 o
. From point B the angle made by the line from B to A, and the line of sight from B to the marker is 53 o . a) How far from A is the marker? b) How far from the marker is the point on AB that is closest to the marker?
Advanced Math II  Unit 5  Triangle Trigonometry 60

Solutions:
1.
Problem # a
1.
2.
2 b c
≈ 1.06 ≈ 2.69
≈ 3.06
2 2
3.
4.
≈ 10.82
4
2
5

A
40 o
100
≈ 11.34 70 none 60 o

40
B
20 o
o

40
C
120
o
5. a
1
≈ 9.07 a
2
≈ 2.20
4
2. 68.3 feet
3. a) ≈ 163.8 meters b) ≈ 161.3 meters
6 A
A
1
2
≈ 129.1
≈ 10.9
10
o
100
o none none
20
o
A
1
≈ 30.9
C
2
≈ 149.1
Activity 4: Using the Law of Cosines (GLEs: 11, 14)
Help students see that the Law of Cosines is used to solve triangles where either 2 sides and the included angle are given (SAS) or 3 sides (SSS). Areas of triangles are also included since the area formula A

1 ab sin C uses the same information used in the Law of Cosines
2
(SAS). When all three sides are given, the student can use Hero’s Formula
Area

(

)(

)(

) where s is the semi-perimeter (one half the perimeter).
1. Use the Laws of Cosines and the area formula to fill in the following table. Have students a) Sketch each triangle b) Show the setup to be used
Problem # a b c

A

B

C Area of the Triangle
1. 8 5 40 o
2. 14 12 120
o
3. 7 13 8
2. A soccer goal is 5 meters wide. When a player is 21 meters from one goal post and 19 meters from the other he shoots for the goal. What is the angle of view of the goal that the player sees as he shoots?
3. Farmer Brown tethers his goat on the longer side of his 20 by 40 foot barn with a rope that is 30 feet long. How many square feet of pasture does the goat have to graze?
Advanced Math II  Unit 5  Triangle Trigonometry 61

Solutions:
Problem
#
1. a b c

A

B

C Area
8 5 ≈ 5.3 ≈ 102.4
o note: using the calculator will give 77.6
o
This is a good time to teach students that sin-1(.97025) has 2 answers;
1 acute and 1 obtuse. Students should find angle B and then use
180 –

A -

C to find

B .
120 o
≈ 37.6 40 o ≈ 12.9
2.
3.
≈
22.54
7
14 12
13 8
≈ 27.8
o
32.5
o 27.5
o ≈
120
o ≈
32.2
o
72.75
≈
24.25
2. 13.2
o
3. 1727.9 ft
2
Activity 5: Navigation Problems using the Laws of Sine and Cosine (GLEs: 11, 14, 16)
Since navigation problems involve both direction and distance, help students understand how to add vectors geometrically in order to set up the problems. A properly drawn diagram

A


B is the vector from the beginning of head of A .

A to the end of

B if the tail of

B is placed at the
The lengths of A, B, and C represent the magnitude of the vector quantity, and the direction of the segment represents the vector’s direction. The navigation problems below will use bearings or headings. A bearing is an angle measured clockwise from north, used universally by navigators for a velocity or displacement vector. For each of the problems, have students set up a diagram such as the one below showing the bearings from the north. The north lines are parallel. Since the properties of parallel lines and the resulting angle relationships enable the student to find the needed angles, review them with the students
For each problem below a) draw and label a diagram
Advanced Math II  Unit 5  Triangle Trigonometry 62

b) use the Laws of Sine and Cosine to find the displacement vector.
1. A ship sails 60 miles on a bearing of 20 o and then turns and sails 40 miles on a bearing of
70
o
. Find the resultant displacement vector as a distance and a bearing.
2. A plane flies 30 miles on a bearing of 220
o
and then turns and flies 40 miles on a bearing of 20 o . Find the resultant displacement vector as a distance and a bearing.
3. A ship proceeds on a bearing of 200
o
for 2 hours at a speed of 18 knots. At that point, it changes course to 250
o
, continuing at 18 knots for 3 more hours. At that time, how far is the ship from its starting point?
4. An airplane is flying at a speed of 500 miles per hour on a heading of 330
o
. It encounters a wind with velocity of 70 miles per hour with a bearing of 315
o
. What is the resultant speed and bearing of the airplane?
5. A bird is starting its annual migration from the marshes of Louisiana. It takes off on a heading of 165
o
with an average speed of 12 miles per hour. On that day there is a 10 mile per hour breeze blowing from the northwest (heading 315
o
) How far has the bird traveled after 8 hours? At what heading has the bird actually traveled?
(There is interesting information on bird migration at http://birding.about.com/library/weekly/aa032898.htm.)
Solutions:
1) ≈ 91 miles bearing 220.7
o
2) ≈ 15.6 miles bearing 261
o
3) ≈ 81.9 nautical miles
4) ≈ 567.9 miles bearing 328.2
o
5) ≈ 170 miles heading 151.4
o
Sample Assessments
General Assessments

The student will complete a writing assessment dealing with some aspect of triangle trigonometry. One example would be to have him/her set up general guidelines for solving trigonometric problems that involve triangles. The teacher will assess student understanding by the student’s using verbs such as show, explain, describe, justify , or compare and contrast.
For instance, the teacher will ask the student to explain how the area formula learned in this unit becomes the familiar area formula for a right triangle.

The student will research one of the many ways that triangle trigonometry is used in the real world. This could include an interview with a professional who uses trigonometry on a regular basis.

The student will also engage in a group activity that will be assessed. The teacher will choose a problem or problems that require more work than the textbook problems.
They can be problems such as those put out by the National Society of Professional
Advanced Math II  Unit 5  Triangle Trigonometry 63

Surveyors in their annual TRIG-STAR contest. This is a contest based on the practical application of Trigonometry. Their website is http://www.acsm.net/trigstar/
Navigation problems make excellent problems. The teacher will give each group a problem which requires them to fly from one city to another on a particular plane with given wind conditions. Each students will have to determine the heading on which to fly as well as the airspeed of that particular plane before solving the problem. Scoring rubric based on
1.
teacher observation of group interaction and work
2.
explanation of each group’s problem to class
3. work handed in by each member of the group
Activity-Specific Assessments

Activity 1
The student will demonstrate proficiency in use of the right triangle ratios. One good way is to give each student an erasable white board and dry erase marker and put the questions on the overhead. The teacher will ask students to work the problems then hold up their white board as soon as they have the answer. Sample problems could include:
1. In triangle ABC B

3 , and side c

15.
Find the length of side b .
5
2.
sec A

13 . Find the other five trigonometric ratios.
12
3. In triangle ABC
 
90

, side a = 4, and side b = 8. Find

A and

B .
4. Geometric shapes containing right triangles that require students to find the missing sides or angles.
5. Trigonometric ratios of the special right triangles (non-calculator)
.

Activities 3 and 4
The student will demonstrate proficiency in using the Sine and Cosine Laws. The teacher will check the student’s ability to determine (a) whether or not the given information will result in a unique triangle and (b) which formula should be used to solve the triangle. Students usually prefer using the Law of Sines. However, the sine ratio is the same for any angle and its supplement, so using the Law of Sines indiscriminately can cause a problem. (See problem #1 in Activity 4)

Activity 5
The student will demonstrate proficiency in using vector addition to find the magnitude and direction of the sum of a vector.
Advanced Math II  Unit 5  Triangle Trigonometry 64