Elizabeth Pawelka Geometric Probability 4/11/12 p.7
Geometry
Lesson Plans
Section 9-3: Angles of Elevation and Depression
4/1712
Warm-up (15 mins)
· Practice 9-1 and 9-2: odd problems
· Challenge: Describe some real-world applications of trigonometry
(astronomy, distances, parallel parking (see http://www.geogebratube.org/student/m3022))
Also: A simple experiment with polarized sunglasses
Suppose one gets two pairs of identicalpolarizedsunglasses (unpolarized sunglasses won't work here), and puts the leftlensof one pair atop the right lens of the other, both aligned identically. If one pair is slowly rotated, the amount oflightthat gets through is observed to decrease until the two lenses are atright anglesto each other, when no light gets through. When the angle through which the one pair is rotated is θ, what fractions of the light that penetrates when the angle is 0, gets through? Answer: it is cos2θ. For example, when the angle is 60 degrees, only 1/4 as much light penetrates the series of two lenses as when the angle is 0 degrees, since the cosine of 60 degrees is ½.
Homework Review (10 mins) – ask for any questions on homework
Homework (H)
· p. 472, # 1-16, 27 - 29
· p. 479, # 1-17, 22 -25
Homework (R)
· p. 472, # 1-16, 28
· p. 479, # 1-17, 24
Statement of Objectives (5 mins)
The student will be able to identify angles of elevation and depression and use them with trigonometric ratios to solve problems.
Teacher Input (55 mins)
Angles of elevation and depression are created by the horizontal lines formed by a person’s line of sight to an object. If a person is looking up, the angle is an elevation angle. If a person is looking down, the angle is a depression angle.
x = angle of elevation from ground to top of tree x = angle of depression from lighthouse to boat
Ask: If you knew how far you were from the tree and the angle of elevation, how could you find the height of the tree? tan(x) = height of tree/distance from tree.
Also ask the students what relationship angles 1 and 2 have to each other. They are congruent b/c they are alternate interior angles.
Examples in Pictures: (in number 1 also ask which angles are congruent)
6.
1. a) angle of depression from top of Bldg B to top of Bldg A
b) angle of elevation from top of Bldg A to top of Bldg B
c) angle of depression from top of Bldg A to bottom of Bldg B
d) angle of elevation from bottom of Bldg B to top of Bldg A
∠1 ≅ ∠2, ∠3 ≅ ∠4
2. tan(53) = x/50 => x = 50 tan(53) = 66.4 ft.
3. tan(7) = x/10,000 => x = 10,000 tan(7) = 1227.8 ft
4. cos(50) = 100/x => x = 100/cos(50) = 155.6 ft
5. sin(40) = 50/x => x = 50/sin(40) = 77.8 ft
6. tan(x) = 10/17.3 => tan-1(10/17.3) = x = 300
Examples in words:
7. A blimp is flying 500 ft above the ground. A person on the ground sees the blimp by looking up at a 250 angle. The person’s eye level is 5 ft above the ground. Find the distance from the blimp to the person to the nearest foot.
sin(25) = 495/x => x = 495/sin(25) = 1171.3 ft
8. A surveyor stands 200 ft from a building to measure its height with a 5 ft tall theodolite. The angle of elevation to the top of the building is 350. How tall is the building?
FYI: Atheodolite(/θiːˈɒdəlaɪt/) is a precision instrument for measuringanglesin the horizontal and vertical planes. Theodolites are mainly used for surveyingapplications,
9. You see a rock climber on a cliff at a 320 angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line-of-sight distance to the rock climber to the nearest tenth of a foot.
cos(32) = 1000/x => x = 1000/cos(32) = 1179.2 ft
10. An airplane flying 3500 ft above ground begins a 20 descent to land at an airport. How many miles from the runway is the airplane when it starts its descent?
11. Two buildings are 30 ft apart. The angle of elevation from top of one to the top of the other is 190. What is their difference in height?
tan(19) = x/30 => x = 30 tan(19) = 10.3 ft
12 In a galaxy far, far away, a spaceship is orbiting the planet Obar. The ship wants to land in a large, flat crater, but the captain of the ship wants to make sure the crater is large enough to hold the ship. When the ship is 4 miles above the planet, the onboard guidance system measures the angles of depression from the ship to both sides of the crater. The angles measure 220 and 370 respectively. What is the distance across the crater? If the spaceship is 2500 ft long, will it fit in the crater?
/ crater = y - xtan(37) = 4/x
x = 4/tan(37)
x = 5.31
tan(22) = 4/y
y = 4/tan(22)
y = 9.9
crater = 9.9 – 5.31 = 4.59 miles
Investigation: What if you didn’t have a theodolite to measure an angle of elevation? What do you need to find an angle? (2 sides of a right triangle.) What if you wanted to find out how high the flag pole is? What could you use? (Hint: on a sunny day … use your height and length of shadow to find angle of elevation of the sun. Then use that angle and the shadow of the object to find the height of the object. Refer to example 6)
Activity and worksheet (if time)
Materials need: tape measures
Activity: (students should be in five groups of about five people)
1. Measure the height of one person in the group.
2. Measure that person’s shadow.
3. Using what you know about trig ratios, determine the angle of elevation from the ground to the sun. (Students should sketch a picture of the situation to help make sense of the computation.)
4. Measure the shadow of the object.
5. Using the angle of elevation and the measure of the shadow, use what you know about trig ratios to determine the height of the object. (Students should sketch a picture of the situation to help make sense of the computation.)
6. Fill in all of the information in your picture (you should now have two right triangles with 2 sides and an angle measure).
Worksheet: (half sheets – 1 per group)
Extension questions/activities:
· Can the sine or cosine of an angle ever be greater than 1? If so, when? If not, why?
· Have the students draw and describe their own application of angles of elevation and depression, complete with a solution.
Closure (5 mins)
· Today you learned to identify angles of elevation and depression and use them with trigonometric ratios to solve problems.
· Tomorrow you’ll learn to use trigonometric ratios to find the area of polygons.
· Quiz tomorrow on Sections 9-1/9-2
Homework (H)
· p. 484, #1 – 21, 23, 28, 33, 34
Homework (R)
· p. 484, #1 – 21, 23, 33