Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Measuring Inaccessible Heights
Angle of
Elevation
An angle of elevation is an angle formed by a ____________________
line (left to right) and a line ______________________ the horizontal.
Angle of
Depression
An angle of depression is an angle formed by a horizontal line
and a line ____________________ the horizontal
Clinometer
A clinometer is a device that is used to __________________________
of elevation, which allows us to measure inaccessible __________
(ie. when the height itself cannot be measured directly).
Clinometers measure the angle _______________________ the angle
of inclination.
To use a clinometer: look through the straw and line up the
object you are measuring in your line of _______________. Have a
partner read the measurement on the protractor as you hold
the straw in place.
Angle of Elevation = 90 – Angle of the Clinometer
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Think
Can you explain why the angle of elevation is ninety minus the
angle of the clinometer? (Hint: triangle angles add up to 180).
Example
Joe is stuck at the bottom of a cliff. He has a clinometer and a
measuring tape with him. He needs to tell the rescue crew how
long the rope will need to be to get him up.
1) He marks where he is standing and measures that his
distance to the cliff is 50m.
2) From the point 50m away from the cliff, he uses the
clinometer to sight the top of the cliff. The clinometer reads
20°, so he knows the angle of elevation is 70°
How can Joe find the height of the cliff?
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Example
The angle on the clinometer reads 50.4° when Khaled, who is
1 km (1000 m) away, looks to the top point of the Burj Khalifa.
He is 1.65 m tall. How tall is the Burj Khalifa?
Practice
The angle on the clinometer when I measure from my eye to
the top of the tree in my backyard is 55°. I am standing 22 feet
away from. I am 5 feet, 6 inches tall. How tall is the tree?
Strategies for Drawing Right Angle Triangle Diagrams
1) Read the question carefully, and more than once if needed.
2) Start by drawing the objects (ie. the tree and person in the
example above), then draw the triangle formed between them.
3) Label the diagram – add all given information.
4) Mark the unknown with an ‘x’ or other variable.
5) Read the question again and check your diagram. Does it
make sense based on the information given in the problem?
6) Keep it simple. Don’t waste time drawing elaborate
pictures; quick sketches and stick people are fine!
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Practice
The angle of elevation from a sailboat to the top of a cliff is 30°.
The sailboat is 300 m away from the base of the cliff. What is
the height of the cliff? Draw a diagram.
Practice
The angle of depression from a cat stuck in a tree to a
firefighter standing on the ground is 40°. The firefighter is
standing 10 meters from the base of the tree. How high is the
cat off of the ground? (Remember: Angle of depression is from
the horizontal downward). Draw a diagram.
Practice
Abdullah is standing 612.5 m from the highest point of Jebel
Hafeet in Al Ain. The angle on the clinometer reads 26°. How
high is Jebel Hafeet? Draw a diagram.
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Making and Using a Clinometer
Activity
Send a representative from your group to collect your
supplies. You will need:
-A protractor
-Straw
-String
-Weight
-Tape (2-3 pieces)
-A measurement assignment
-White paper
-Measuring tape
Create your clinometer by following these steps:
1) Tie the weight to one end of the string. Tie the other end of
the string to the middle of the straw.
2) Tape the straw to the front of the protractor along the
straight bottom. Make sure that it is lined up perfectly with the
horizontal line on the protractor.
3) Slide the string so that is lines up with the origin of the
protractor (the middle point of the bottom of the protractor).
Think
How will you use the clinometer to measure your assigned
item? Discuss in your group and record below.
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
When you have recorded your strategy, have Ms. Courtney
check your work. Then use your clinometer and measuring
tape to gather the necessary data to find the unknown height.
Calculation
Use the space below to calculate the height of the object.
Activity
Create a mini-poster (use an 8x11 sheet of white paper) to
summarize your work. Include:
1) A title, which includes the object you measured.
2) A diagram
3) A written description of how you gathered your data
4) Your calculations
5) A conclusion
6) Your names
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Problems Involving More than One Right Triangle
In some applications of trig. ratios, more than one triangle is
used to solve the problem. The steps below will help to guide
you:
Note
1) Draw a diagram if one is not given.
2) Label the diagram with all given information.
3) Re-read the question and make sure that your diagram
matches the question.
4) List the information you are being asked to find.
5) If it is not clear how to find the answer, list the information
that you can find and start there. You may have to find one of
these measurements before you can find the measurement the
question asks for.
Example
Find the unknown measurement below.
Think
What measurements can we calculate based on the given
information?
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Applications of Trig. Ratios
Math 10 Plus Notes
Practice
Date:
a) Calculate the length of x.
b) Calculate the length of side CD.
Example
From the top of a 20m building, a surveyor measured the
angle of elevation to the top of another building and the
angle of depression to the base of the other building. Find the
height of the taller building.
We can’t calculate PR in one step because it is not a side
of a right angle triangle. However, we know that SR is 20
m because it is the distance from the ground to the top
of the smaller building. We now need to calculate PS:
The height of the taller building is _______ + _______ = ________ m.
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Practice
Find the height of the taller building.
Practice
A tourist is standing at the top of the Peggy’s Cove lighthouse.
He sees a fishing boat at an angle of depression of 23° and a
sailboat at an angle of depression of 9°. If the tourist is 33.5 m
above the water, determine the distance between the boats.
Practice
If PQ = 8.2 cm, QS = 5.3 cm and ST = 7.3 cm, find RS.
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
3D Trigonometric Ratio Problems
Note
You can use a cue card to create a 3D model which will help
you visualize the problem.
-Fold along the dotted lines, as shown.
-Cut along one dotted line from the edge to the center.
-Fold to create the corner of a box. The lengths and
angles can then be drawn on the card.
Example
Create a 3D model for the triangles shown below. Find the
length of x.
Practice
The lengths of CO and OB are equal. Find the length of CB
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Applications of Trig. Ratios
Math 10 Plus Notes
Date:
Example
From the top of a 90 ft. observation tower, a fire ranger
observes a fire to the west at an angle of depression of 5°. He
observes another fire south of the tower at an angle of
depression of 2°. How far apart are the fires?
Practice
A communication tower is 35 m tall. North of the tower,
Miriam measures the angle of elevation to the top of the tower
as 70°. East of the tower, Sarah measures the angle of
elevation to the top of the tower as 50°. How far apart are
Miriam and Sarah?
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