CC Geometry H
Aim #26: Students solve problems using the tangent ratio.
Do Now: a. Use a calculator to find tangent θ, and sin/cos θ. Give your answer
rounded to the ten-thousandth place. θ
0
10
20
30
40
50
sin θ
0
0.1736 0.3420
cos θ
1
0.9848 0.9397 0.8660 0.7660 0.6428
0.5
60
70
80
0.6428 0.7660 0.8660 0.9397 0.9848
0.5
0.3420 0.1736
90
1
0
sin θ
cos θ
tan θ
b. What do you notice about the numbers in the row sin θ/cos θ compared with the
numbers in the row tan θ?
• How would you describe the angle of elevation shown below?
• How would you describe the angle of depression?
The angle of elevation
or depression is the
angle between the
horizontal and the line
of sight.
• What do you notice about the measure of the angle of depression and the
angle of elevation? Why?
1) Scott, whose eye level is 1.5 m above the ground, stands 30 m from a tree. The
angle of elevation of a bird at the top of the tree is 360. How far above the
ground is the bird?
2) Standing on his rooftop, John watches his friend approach his building. John's
angle of depression is 400. The rooftop is 16 m above the ground, and John's eye
level is 1.8 m above the rooftop. What is the distance between John's friend and
the building?
3) Standing on the deck at the top of a lighthouse, a person spots a ship at an angle
of depression of 200. The lighthouse is 28 m tall and sits on a cliff 45 m tall as
measured from sea level. What is the horizontal distance between the lighthouse
and the ship? Sketch a diagram and show work.
4) Samuel is at the top of a tower and will ride a trolley down a zip-line to a lower
tower. The total vertical drop of the zip-line is 40 ft. The zip-line's angle of
elevation from the lower tower is 11.50. What is the horizontal distance between
the towers?
How is the slope of lines AB and CD related to the tangent of the triangle formed
by each line with the x and y axis?
B
A
angle of
elevation
C
angle of depression
D
We note that the tangent ratio =
Explain.
5) A line on the coordinate plane makes an angle of elevation of 530. Find the slope
of the line, correct to four decimal places.
530
6) A line on the coordinate plane makes an angle of depression of 360. Find the
slope of the line, correct to four decimal places.
7) Given the angles of depression below, determine the slope of the line with the
indicated angle correct to four decimal places.
a) 350 angle of depression
d) 870 angle of depression
b) 490 angle of depression
e) 890 angle of depression
c) 800 angle of depression
f) 89.90 angle of depression
g) What appears to be happening to the slopes (and tangent values) as the angles
of depression get closer to 900?
h) Find the slopes of angles of depression that are even closer to 900 than 89.90.
Can the value of the tangent of 900 be defined? Why or why not?
8) The pitch of a roof on a home is expressed as a ratio of
vertical rise : horizontal run where the run has a length of 12 units. If a given
roof design includes an angle of elevation of 22.50, and the roof spans 36 ft. as
shown in the diagram, determine the pitch of the roof (a ratio). Then determine
the distance along one of the two sloped surfaces of the roof, to the nearest
tenth.
Name_____________________
Date _____________________
CC Geometry H
HW #26
1) The line on the coordinate plane makes an angle of depression of 240. Find the
slope of the line, correct to four decimal places.
2) For each indicated acute angle below, express the sine, cosine, and tangent
ratios in simplest radical form.
3) An anchor cable supports a vertical utility pole forming a 510 angle with the
ground. The cable is attached to the top of the pole. If the distance from the
base of the pole to the base of the cable is 5 meters, how tall is the pole, to the
nearest hundredth?
4) A ship is heading toward a lighthouse whose beacon is 120 ft about sea level.
At a sighting from point A, the angle of elevation from the ship to the light was
90. A while later, at point F, the angle was 200. To the nearest foot, determine
how far the ship traveled from A to F.
A
F
Review:
1. Use the Pythagorean Theorem to find the missing side length in simplest form:
2
2. Find the exact value of x and y.
10
y
x
20
3. Use sin and cos to find the missing side length in simplest form:
y
x
300
12
4. Find the exact length of the altitude and the exact area of an equilateral
triangle with side 28.
5. Write an algebraic expression equivalent to sin x when 0o < x < 90o.