The environmental club at your school has taken on the task of planting a garden to provide fresh fruit, herbs, and vegetables to the school cafeteria. The approved area has several trees that need to be moved and replanted so that you can use the garden space more efficiently. A building supply company donates several pieces of wire, each 13 feet in length, to help brace the replanted trees.
As the president of the environmental club, you must provide the specifications to the club members and volunteers before they can begin the replanting process. At most, the ladders you have can reach 11 feet 3 inches up the trunk of a tree. Given this information, what angle should the wires make with the ground, and at what horizontal distance from the tree trunk should the wires be staked to the ground?
13 feet
11 feet 3 inches
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
Coaching
a. What is the maximum height, in inches, each wire can be secured to a tree?
b. What is the length of each wire in inches?
c. Based on the given information, which trigonometric function is required to find the angle at which each wire should be staked to the ground? What is the angle measure?
d. What right triangle property should you use to most accurately find the distance from the trunk of the tree to the spot where the wire will be staked to the ground?
e. What is the distance, in inches, from the trunk of the tree to the spot where the wire will be staked to the ground?
Problem-Based Task 2.2.1: Replanting Project
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
Coaching Sample Responses
a. What is the maximum height, in inches, each wire can be secured to a tree?
The wires can be secured as high up as the ladders will reach. Since this is known to be 11 feet 3 inches, start by converting 11 feet to inches.
(11 ft) 12 in
Add 3 inches to determine the total height in inches.
132 + 3 = 135
The maximum height each wire can be secured is 135 inches.
b. What is the length of each wire in inches?
Convert the known length of the wires (13 feet) to inches.
(13 ft) 12 in
c. Based on the given information, which trigonometric function is required to find the angle at which each wire should be staked to the ground? What is the angle measure?
G
156 in
135 in
The opposite side of the angle and the hypotenuse are known, so sine should be used.
sin G =
G = 59.926º. The angle of elevation is approximately 60º.
Problem-Based Task 2.2.1: Replanting Project
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
d. What right triangle property should you use to most accurately find the distance from the trunk of the tree to the spot where the wire will be staked to the ground?
Since two side lengths were given, you could use either trigonometry or the Pythagorean Theorem. However, in an effort to be as precise as possible, the best choice is the Pythagorean Theorem. With trigonometry, you would have to use the acute angle G that we just
approximated in part c.
e. What is the distance, in inches, from the trunk of the tree to the spot where the wire will be staked to the ground?
From part d we determined that the Pythagorean Theorem is the most accurate method for determining the distance.
d = unknown leg = 135 in
hypotenuse = 156 in d 2 + 1352 = 1562 d 2 + 18,225 = 24,336 d 2 = 6111
d = t 6111 d = 78.173
The distance from the tree trunk to the staked wire is about 78 inches.
Recommended Closure Activity
Select one or more of the essential questions for a class discussion or as a journal entry prompt.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
continued
Draw a sketch of the triangle relationship for all problems that do not have a diagram. Unless otherwise stated, round all side lengths to the nearest thousandth and round angle measures to the nearest degree.
1. In -TRY , Y is a right angle and sin R = 3
5. What are the cosine and tangent of T ? Write your answers as fractions and as decimals.
2. In -RAY , Y is a right angle and tan R = 11
60. What are the cosine and sine of A ? Write your answers as fractions and as decimals.
3. A ladder manufacturer recommends that its ladders be used on level ground at an angle of 72.5º to the horizontal. At that angle, how far up the side of a building will the top of a 14-foot ladder reach?
4. At a certain time of day, an 18-foot pole casts a shadow that forms an angle of 50º with the ground. How long is the shadow?
5. A right triangle has legs that each measure 6 inches. Use a trigonometric function to find the measure of the triangle’s acute angles to the nearest degree. Why does the solution make sense in this situation?
Practice 2.2.1: Calculating Sine, Cosine, and Tangent
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
6. What is the value of x?
0.34 0.15
x˚
7. One section of a ski run is 650 feet long and falls 260 feet in elevation at a constant slope. To the nearest degree, what angle does the ski run form with the horizontal?
650 ft 260 ft
θ
8. Solve the right triangle below.
58˚
32
B
C A
continued
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
9. Solve the right triangle XYZ, where Y is a right angle, XZ = 12 inches, and XY = 8 inches.
10. A photographer shines a camera light at a particular painting, forming an angle of 47º from the camera’s horizontal line of sight. If the light is 52 feet from the wall where the painting hangs and the camera lens is 5 feet from the floor, how high above the floor is the painting?
x
52 feet
5 feet
47°
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
Your class is reviewing for the midterm exam with a series of activities. One activity features a bag with slips of paper folded separately inside it. Each slip of paper has a different side length of the right triangle below written on it. Your directions are to pull two numbers from the bag and set up a ratio, using the first number as a numerator and the second number as a denominator.
3
4 5
C A
B
1. What is the probability that, given the three basic trigonometric ratios, the ratio of the numbers will be the cosine of B ? Explain.
2. What is the measure of B to the nearest degree? Explain.
Lesson 2.2.2: Calculating Cosecant, Secant, and Cotangent
Warm-Up 2.2.2
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
Common Core Georgia Performance Standard MCC9–12.G.SRT.8
Lesson 2.2.2: Calculating Cosecant, Secant, and Cotangent
Your class is reviewing for the midterm exam with a series of activities. One activity features a bag with slips of paper folded separately inside it. Each slip of paper has a different side length of the right triangle below written on it. The directions are to pull two numbers from the bag and set up a ratio, using the first number as a numerator and the second number as a denominator.
3
4 5
C A
B
1. What is the probability that, given the three basic trigonometric ratios, the ratio of the numbers will be the cosine of B ? Explain.
In order to determine the probability of pulling out the numbers that create the cosine of B , you must first know all of the possible ratios.
sin
There are six trigonometric ratios possible, and one of them is the cosine of B . As such, the probability of pulling the cosine of B is 1
6.