Applied Trigonometry
Geometry Honors 7.4
Steps to Solving Real-World Problems with Trigonometry:
Step 1: After reading the problem, underline all lengths and angles
Step 2: Draw a diagram (if there isn’t one!)
Step 3: Write key lengths or angles into diagram
Step 4: Determine if we use sine, cosine or tangent to solve
Step 5: Set up an equation to solve
Step 6: Check your answer, does it make sense?
Warm up: Let’s start with a basic problem similar to the problems from 7.3…
1. While walking, Rafael and his dog Rocky see the Eiffel Tower and notice that the angle of elevation
is 23⁰. If Rafael is 2,220 feet from the middle of the base of the arch, how tall is the Eiffel tower?
Angle of Elevation and Depression
Another way to describe the location of angles in real-world right
triangle problems is the angle of elevation (when looking up at an
object) and the angle of depression (when looking down at an
object). Both are the angle found between the horizontal line of
sight and the segment connecting the two objects.
What do you think is true about the relationship between the angle
of depression and the angle of elevation? Explain.
Ben is on the diving board at the neighborhood pool, Jenna is in the
pool, and a lifeguard sits at her station on the opposite end of the pool.
Classify each angle as an angle of elevation or an angle of depression.
1___________
2 __________
3 ___________
4 __________
[1]
1. A man standing on level ground is 1000 feet away from the base of a 350-foot-tall building. Find, to the
nearest degree, the measure of the angle of elevation to the top of the building from the point on the
ground where the man is standing.
2.
A ship is using sonar to find sunken treasure on the ocean floor. The angle of depression of the sonar to
the bottom of the ocean is 30°. The distance from the ship straight down to the bottom of the ocean is 250
feet. How far is the ship from the sunken treasure?
3. A man standing on a cliff is watching a boat sailing on the ocean. The man estimates the boat is 100 feet
from the bottom of the cliff. The man knows the cliff is 50 feet tall. What is the angle of depression?
4.
A steel cable extends from the top of a building to a point on the ground that is 1000 ft from the base
of the building. At the point where the cable is anchored to the ground, it is determined that the
measure of the angle of elevation is 42°. To the nearest foot, how tall is the building?
[2]
5. Rose is flying a kite and has played out 300 feet of string. The kite is 120 feet above the ground and Rose is
5 feet tall. To the nearest tenth of a degree, at what angle of elevation does she sight the kite?
6. An airplane pilot can see the top of a traffic control tower at a 20° angle of depression. The straight-line
distance between the plane and the top of the tower is 5,000 feet. To the nearest foot, how far above the
tower is the plane?
Area of Triangle
A sail maker is designing a sail with the dimensions given in the diagram.
How can we figure out the amount of fabric that is needed?
[3]
Area of a Triangle
Although all the formulas work the most common representation is A 
1. Find the area of each triangle.
a.
1
ab sin C
2
b.
2. Find the area of ∆ ABD to the nearest square unit.
B
25
A
70
D° 5
C
4. Find the exact area of a triangle if two sides measure 12 in and 15 in with an included angle of 60⁰.
[4]
Name ______________________________________
Geometry Honors
7.4 ws
1. A forest ranger looking out from a ranger’s station can see a forest fire at a 35° angle of depression. The
ranger’s position is 100 ft above the ground. To the nearest foot, how far is it from the base of the ranger’s
station to the fire on level ground with it?
2. The Occupational Safety and Health Administration (OSHA) provides standards for safety at the workplace.
A ladder is leaned against a vertical wall at an angle of elevation of approximately 75 . If the ladder is 25 ft.
long, what is the distance from the base of the ladder to the base of the wall? Round your answer to the
nearest tenth. To the nearest tenth, how high on the wall does the ladder make contact?
3. 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 36 . How far above the ground
is the bird? Round your answer to the nearest tenth of a meter.
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x
x
to find BC. Jill uses the equation cos 41 
to find BC.
16
16
Who is correct? Explain your reasoning.
4. Julie uses the equation sin 49 
5. Samuel is at the top of a tower and will ride 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 11.5 . To the
nearest hundredth, how long is the zip line?
6. Standing on the deck of a lighthouse a person spots a ship
at an angle of depression of 20 . 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? Round your answer to the nearest whole meter.
7. Your family room has a sliding glass door. You want to buy an awning that will be
just long enough to keep the sun out when it is at its highest point in the sky.
The angle of elevation of the rays of the sun at this point is 70⁰, and the height of
the door is 8 feet. Your sister claims you can determine how far the overhang
should extend by multiplying 8 by tan 70⁰. If she is correct, show the set up of the
equation. If she is incorrect set up and solve the correct equation and explain her mistake.
[6]
Area of Triangle: A = ½ ab sinC
7. Find the area of ∆𝐴𝐵𝐷 to the nearest square unit.
B
25
35
A
50°
D 7
C
8. Beatrice says the area of the triangle below is 40 square units because it has a base of 10 units and a
height of 8 units. Do you agree with Beatrice? Why or why not?
10
110°
8
9. A landscaper was called to bring topsoil for customers house to cover their entire garden which
has the following dimensions. Each 40 pound bag of topsoil will cover approximately
.75 square yards and costs $3.09. (This is a great example of a 6-point regents question)
a. Find the cost of the topsoil to the landscaper.
b. If the landscaper wants to earn a profit of 40%, how much should they charge their customer?
[7]