A high school biology class is planning a vegetable garden. The garden will be on the west side of the school building. The students are concerned that the garden will not get enough direct sunlight in the morning if the building’s shadow shades the garden too much. Starting May 1, the garden needs direct sunlight by 10 A.M. at the latest. The school building is 8 meters high. The garden is 6 meters from the building. The students find on the Internet that the angle of the sun’s rays at 10 A.M. on May 1 will be 60º above the horizon. How long will the building’s shadow be? Is the chosen location a good spot for the garden? What are the sine of 30º and the cosine of 60º?
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios
NAME:
Coaching
a. Draw a picture of the problem.
b. What are the measures of the angles in the drawing’s triangle?
c. What is the ratio between the shortest side length and the hypotenuse in this triangle?
d. What is the length of the hypotenuse if the shortest side length of this triangle is 1 inch?
e. Find the length of the longer side by using the Pythagorean Theorem.
f. What are the basic side lengths of such a triangle if the shortest side length is 1 unit?
g. What is the scale factor of the triangle in the drawing?
h. How can you use the scale factor to find the length of the building’s shadow?
i. Is the chosen location a good spot for the garden? Explain.
j. What are the sine of 30º and the cosine of 60º?
Problem-Based Task 2.1.2: School Vegetable Garden
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios
Instruction
Coaching Sample Responses
a. Draw a picture of the problem.
School
8 m
6 m
Garden 60˚
b. What are the measures of the angles in the drawing’s triangle?
The angles measure 30º, 60º, and 90º.
c. What is the ratio between the shortest side length and the hypotenuse in this triangle?
Draw a triangle that has angles of 30º, 60º, and 90º.
Measure the shortest side length, a.
Measure the hypotenuse, c.
Divide the hypotenuse length by the shortest side length, a.
The ratio is 2 : 1 (or 1 : 2). The hypotenuse is twice as long as the shortest leg.
d. What is the length of the hypotenuse if the shortest side length of this triangle is 1 inch?
Draw a 30º–60º–90º triangle so that the shortest side length = 1 inch.
The length of the hypotenuse = 2 inches.
e. Find the length of the longer side by using the Pythagorean Theorem.
Substitute values for a and c.
12 + b2 = 22 b2 = 3
b = 3 inches
Problem-Based Task 2.1.2: School Vegetable Garden
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios
Instruction
f. What are the basic side lengths of a 30º–60º–90º triangle if the shortest side length is 1 unit?
The basic proportional lengths opposite the angles in a 30º–60º–90º triangle are:
opposite side length of 30º angle (shortest side length): 1 opposite side length of 60º angle (longest side length): 3 opposite side length of 90º angle (hypotenuse): 2
g. What is the scale factor, k, of the 30º–60º–90º triangle in the drawing?
The only known side length (the longer side opposite the 60º angle) is 8 meters.
The basic length of the longer side in a 30º–60º–90º triangle is 3 . 3 •k"8 meters
8
" 3 k
k ~ 4.619 meters
h. How can you use the scale factor to find the length of the building’s shadow?
The shadow is opposite the 30º angle in the drawing. This makes it the shortest side length.
The basic length of the shortest side in a 30º–60º–90º triangle is 1.
Multiply the scale factor by the basic side length of the triangle that corresponds with the building’s shadow. This is the short side of the triangle, and the basic length is 1. Therefore, the length of the building’s shadow is approximately 4.619 meters.
i. Is the chosen location a good spot for the garden? Explain.
Yes, because the garden will not have shade at 10 A.M. on May 1. Since the garden will be placed 6 meters from the building, and the building’s shadow will be about 4.619 meters long, the garden will be outside of the shade.
j. What are the sine of 30º and the cosine of 60º?
sin 30º = 0.5 cos 60º = 0.5
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 1: Exploring Trigonometric Ratios
NAME:
continued
Use what you have learned about the sine and cosine identities to complete the following problems.
1. sin 40º ~ 0.643. What is cos 50º?
2. cos 25º ~ 0.906. Find the sine of the complementary angle.
3. Find a value of V for which sin V = cos 22º is true.
4. Find a value of V for which cos V = sin 59º is true.
5. sin 5º ~ 0.087. Use this information to write an equation for the cosine.
6. cos 34º ~ 0.829. Use this information to write an equation for the sine.
Practice 2.1.2: Exploring Sine and Cosine As Complements
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios
NAME:
7. sin 20º + sin 10º ~ 0.517. Find V if cos V + cos 80º ~ 0.517.
8. Use the diagrams of -ABC and -DEF to fill in the empty boxes for the following equations.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios
NAME:
9. Lucia needs to buy a ladder to reach her roof. The edge of the roof is 21.125 feet high. To use a ladder safely, the base of a ladder should be 1 foot away from a building for every 3.25 feet of building height. The ladder should extend 3 feet longer than the edge of the roof. How long does the ladder need to be?
10. Use the information you found in problem 9 to draw the triangle created by leaning the ladder against the house at the recommended distance. Label the angles so that A is where the ladder meets the building, B is where the ladder meets the ground, and C is where the building meets the ground. Then, calculate the sine and cosine values for angles A and B.