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Created by Ethan Fahy
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NCTM: Use trigonometric relationships to
determine lengths and angle measures.
NCTM: Use geometric ideas to solve problems
in, and gain insights into, other disciplines
and other areas of interest such as art and
architecture.
Michigan Department of Education:
Differentiate and analyze classes of functions
including linear, power, quadratic,
exponential, and circular and trigonometric
functions, and realize that many different
situations can be modeled by a particular
type of function.
Michigan Department of Education: Use
proportional reasoning and indirect
measurements, including applications of
trigonometric ratios, to measure inaccessible
distances and to determine derived measures
such as density.
Back
Find the sine, cosine, and tangent of an acute
angle within a right triangle.
Use trigonometric ratios to find side lengths in
right triangles
To apply trigonometric concepts in order to solve
real-world problems.
Next
How it Works: This activity will introduce the three geometric ratios
(sine, cosine, tangent) and teach the relationships through a series of
examples and activities. As you proceed through the PowerPoint you
will have an opportunity to learn these concepts. There are tutorials,
guided examples, and links to additional resources which should all
be used before proceeding to the quiz. The quiz portion has three
different difficulty levels (Beginner, Moderate, Expert). Once you can
complete all three levels you will have a thorough understanding of
these concepts.
Audience: The intended audience
is for any student enrolled in high
school geometry.
Key: During the slideshow the
following links will always:
Take you to the Home Page
Take you to the Quiz Page
Back
Goal Of Learner: Recognize
Geometric patterns and be able to
apply them to right triangles and
solve for missing sides.
Next
The home page is the learning center for this activity. Each of the following
links will take you to an informational page where you see definitions,
examples, and applications to help you better understand the material. You
will also find many links that will take you to additional resources to deepen
your understanding. When you have completed each link click on the Quiz link.
Vocabulary:
Definitions of trigonometric
ratios
Finding Trig Ratios:
Calculations, Examples , and
Diagrams
Special Right
Triangles:
Click here if you
need a calculator
Examples and Diagrams
Link to:
Multilingual
Glossary
Solving Right
Triangles:
Examples and Diagrams
Calculating
Trigonometric Ratios
Back
Quiz
Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will
study. If you struggle remembering the ratios use this mnemonic device. Soh – Cah – Toa.
Sin =
Opposite
Hypotenuse
Cos =
Definition
Adjacent
Hypotenuse
Tan =
Opposite
Adjacent
Symbols
Sine: Ratio of the length of the leg
opposite the angle to the length of
the hypotenuse
Notation Used: Sin
Cosine: Ratio of the length of the
leg adjacent the angle to the length
of the hypotenuse
Notation Used: Cos
Tangent: Ratio of the length of the
leg opposite the angle to the length
f the leg adjacent to the angle.
Notation Used: Tan
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Need more
Explanation
By the AA Similarity Postulate, a right triangle with a given acute angle is similar to
every other right triangle with that same acute angle measure. So ∆ABC ~ ∆DEF ~
∆XYZ, and
. These are trigonometric ratios. A trigonometric ratio is
a ratio of two sides of a right triangle.
Back
Home
Directions: Find Sin 30o
1. Identify 30o.
2. Label the sides of the triangle with
relationship to 30o.
3. Use the lengths to find the ratio
opposite
(across from angle)
adjacent
(side touching the angle)
Link to:
Guided Note Sheet
In trigonometry, the letter of
the vertex of the angle is
often used to represent the
measure of that angle.
For example: Sine of A is
written as SinA.
Home
In trigonometry, we often use special right triangles to find exact values of
trigonometry functions.
The key to accomplishing this task is to
1. Recall each reference triangle and accurately
construct it.
2. Identify the angle measure you will be using.
3. Recall the trigonometry ratio and fill in the
values.
45-45-90
30-60-90
Home
Caution:
Do not round answers
if the directions ask for
exact solutions.
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Guided Example: Find the Length of BC
(round your answer to the nearest tenth)
opposite
adjacent
Caution:
Do not round answers until
the final step of your answer.
Use the values of the
trigonometric ratios provided
by your calculator.
Home
Tan15o
BC =
10.2 ft
Tan15o
=
AC
BC
= 38.1
=
10.2 ft
BC
To calculate a trigonometric ratio you just have to plug it into the calculator accurately.
Caution:
Be sure your calculator is
in degree mode, not radian
mode when computing
trigonometric functions. On
a graphing calculator it is
under the mode button.
Guided Ratio: Use your calculator to find
the trigonometric ratio. Round to the
nearest hundredth.
cos76o
sin8o
Cos(76)
Sin(8)
.2419218956
cos(76o)=0.24
Home
.139173101
sin(8o)=7.12
Beginner
Click here if you
need a calculator
Moderate
Link to:
Multilingual
Glossary
Home
Expert
Question 1
Question 3
Question 2
Move on to
Moderate Quiz
Home
Quiz
What is the ratio for cos (cosine)?
Opposite
Hypotenuse
Back
Adjacent
Hypotenuse
Hypotenuse
Opposite
Quiz
What is the ratio for cos (cosine)?
Remember what Soh - Cah – Toa represents.
Help
Back
What is the ratio for cos (cosine)?
Recall:
Sin =
Help
Back
Opposite
Hypotenuse
Cos =
Tan =
Opposite
Adjacent
Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will
study. If you struggle remembering the ratios use this mnemonic device. Soh – Cah – Toa.
Sin =
Opposite
Hypotenuse
Cos =
Definition
Sine: Ratio of the length of the leg
opposite the angle to the length of
the hypotenuse
Notation Used: Sin
Cosine: Ratio of the length of the
leg adjacent the angle to the length
of the hypotenuse
Notation Used: Cos
Tangent: Ratio of the length of the
leg opposite the angle to the length
f the leg adjacent to the angle.
Notation Used: Tan
Back
Adjacent
Hypotenuse
Symbols
Tan =
Opposite
Adjacent
You are ready to move on!!
Quiz
Next
Which trigonometric function would you use to find
Sin
Back
Cos
Opposite
Adjacent
Tan
Quiz
Which trigonometric function would you use to find
Remember what Soh - Cah – Toa represents.
Help
Back
Opposite
Adjacent
Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will
study. If you struggle remembering the ratios use this mnemonic device. Soh – Cah – Toa.
Sin =
Opposite
Hypotenuse
Cos =
Definition
Sine: Ratio of the length of the leg
opposite the angle to the length of
the hypotenuse
Notation Used: Sin
Cosine: Ratio of the length of the
leg adjacent the angle to the length
of the hypotenuse
Notation Used: Cos
Tangent: Ratio of the length of the
leg opposite the angle to the length
f the leg adjacent to the angle.
Notation Used: Tan
Back
Adjacent
Hypotenuse
Symbols
Tan =
Opposite
Adjacent
You are ready to move on!!
Quiz
Next
Write the trigonometric ratio of
sinJ as a fraction.
11
61
Back
60
11
60
61
Quiz
Write the trigonometric ratio of
sinJ as a fraction.
adjacent
opposite
Be sure to follow the steps and
accurately label the sides of the
triangle.
Help
Back
Write the trigonometric ratio of
sinJ as a fraction.
adjacent
Recall:
opposite
Sin =
Help
Back
Opposite
Hypotenuse
Trigonometric Ratio is a ratio of two sides of a right triangle. There are three ratios that we will
study. If you struggle remembering the ratios use this mnemonic device. Soh – Cah – Toa.
Sin =
Opposite
Hypotenuse
Cos =
Definition
Sine: Ratio of the length of the leg
opposite the angle to the length of
the hypotenuse
Notation Used: Sin
Cosine: Ratio of the length of the
leg adjacent the angle to the length
of the hypotenuse
Notation Used: Cos
Tangent: Ratio of the length of the
leg opposite the angle to the length
f the leg adjacent to the angle.
Notation Used: Tan
Back
Adjacent
Hypotenuse
Symbols
Tan =
Opposite
Adjacent
You are ready to move on!!
Quiz
Next
Question 1
Question 3
Question 2
Move on to
Expert Quiz
Home
Quiz
Calculate sin35o. Round your answer to the nearest hundredth.
Error
Back
0.57
-0.43
Quiz
Calculate sin35o. Round your answer to the nearest hundredth.
Be sure you are typing in sin, cos, tan and NOT sin-1, cos-1, tan-1. The -1 represents
the inverse function which is not what you have learned yet.
Help
Back
Calculate sin35o. Round your answer to the nearest hundredth.
Be sure you calculator is in degrees not radians. You need to check this every
time you reset the memory on your calculator.
Help
Back
To calculate a trigonometric ratio you just have to plug it into the calculator accurately.
Caution:
Be sure your calculator is
in degree mode, not radian
mode when computing
trigonometric functions. On
a graphing calculator it is
under the mode button.
Guided Ratio: Use your calculator to find
the trigonometric ratio. Round to the
nearest hundredth.
cos76o
sin8o
Cos(76)
Sin(8)
.2419218956
cos(76o)=0.24
Back
.139173101
sin(8o)=7.12
You are ready to move on!!
Quiz
Next
Given the following diagram:
Find the cosB and write your answer as a
decimal rounded to the nearest hundredth.
0.28
Back
3.57
0.29
Quiz
opposite
Given the following diagram:
adjacent
Find the cosB and write your answer as a
decimal rounded to the nearest hundredth.
Be sure to follow the steps and
accurately label the sides of the
triangle.
Help
Back
Given the following diagram:
Find the cosB and write your answer as a
decimal rounded to the nearest hundredth.
The hypotenuse is always the longest side of a right triangle. So the denominator
of a sine or cosine ratio is always greater than the numerator. Therefore the
sine and cosine of an acute angle are always positive numbers less
than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs,
it can have any value greater than 0.
Help
Back
Directions: Find Sin 30o
1. Identify 30o.
2. Label the sides of the triangle with
relationship to 30o.
3. Use the lengths to find the ratio
In trigonometry, the letter of
the vertex of the angle is
often used to represent the
measure of that angle.
For example: Sine of A is
written as SinA.
opposite
(across from angle)
Link to:
Guided Note Sheet
adjacent
(side touching the angle)
Back
You are ready to move on!!
Quiz
Next
Use a special right triangle to write cos30° as a fraction.
(Do not round your answer)
1
2
Back
√1
3
√3
2
Quiz
Use a special right triangle to write cos30° as a fraction.
(Do not round your answer)
Be sure that you construct the
correct special right triangle.
Help
Back
30-60-90
Use a special right triangle to write cos30° as a fraction.
(Do not round your answer)
Be sure you are using the appropriate
relationship for cos (cosine):
Cos =
Help
Back
Adjacent
Hypotenuse
In trigonometry, we often use special right triangles to find exact values of
trigonometry functions.
The key to accomplishing this task is to
1. Recall each reference triangle and accurately
construct it.
2. Identify the angle measure you will be using.
3. Recall the trigonometry ratio and fill in the
values.
45-45-90
Back
30-60-90
Caution:
Do not round answers
if the directions ask for
exact solutions.
You are ready to move on!!
Quiz
Next
Question 1
Question 3
Question 2
Home
Quiz
Find the length of QR. Round to the
nearest hundredth.
11.49
cm
Back
2.16
cm
5.86
cm
Quiz
Find the length of QR. Round to the
nearest hundredth.
Be sure you are using the appropriate
steps:
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Help
Back
Find the length of QR. Round to the
nearest hundredth.
Be sure you are using the appropriate
steps:
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Help
Back
X
hypotenuse
(always across from right angle)
____63o =
X
12.9 cm
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Guided Example: Find the Length of BC
(round your answer to the nearest tenth)
opposite
adjacent
Caution:
Do not round answers until
the final step of your answer.
Use the values of the
trigonometric ratios provided
by your calculator.
Back
Tan15o
BC =
10.2 ft
Tan15o
=
AC
BC
= 38.1
=
10.2 ft
BC
You have almost completed all the quizzes!!
Quiz
Next
Find the length ED. Round to the nearest
hundredth.
72.29
cm
Back
16.20
cm
15.54
cm
Quiz
Find the length ED. Round to the nearest
hundredth.
Caution:
Be sure your calculator is in degree mode,
not radian mode when computing
trigonometric functions. On a graphing
calculator it is under the mode button.
Make sure your measurements make
sense in relationship to other sides of the
triangle.
Help
Back
Find the length ED. Round to the nearest
hundredth.
Be sure you are using the appropriate
steps:
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Help
Back
opposite
X
adjacent
tan 39o =
To find a length of a right triangle
just follow these simple steps:
1. Identify the angle you are using
2. Label the sides of the triangle
3. Set up your ratio
4. Solve for the missing side
Guided Example: Find the Length of BC
(round your answer to the nearest tenth)
opposite
adjacent
Caution:
Do not round answers until
the final step of your answer.
Use the values of the
trigonometric ratios provided
by your calculator.
Back
Tan15o
BC =
10.2 ft
Tan15o
=
AC
BC
= 38.1
=
10.2 ft
BC
One more question to go!!
Quiz
Next
You are a contractor building a wheelchair ramp to replace stairs for one of the
doorways into the school. The door is 1.8 feet above the ground. To meet the
Americans with Disabilities Act the ramp from the ground to the door must be
at a 2.86o with the ground. To the nearest hundredth of a foot, what is the horizontal
Distance covered by the ramp on the ground?
36.08
feet
Back
6.22
feet
36.03
feet
Quiz
You are a contractor building a wheelchair ramp to replace stairs for one of the
doorways into the school. The door is 1.8 feet above the ground. To meet the
Americans with Disabilities Act the ramp from the ground to the door must be
at a 2.86o with the ground. To the nearest hundredth of a foot, what is the horizontal
Distance covered by the ramp on the ground?
Caution:
Be sure that you draw your diagram and label
it accurately. This will help identify what you
already know and what you are solving for.
Be sure to fill in the numerical values.
Help
Back
Username: mrfahy
Password: geometry
Page 528
Example 5
Ramp
Height
You are a contractor building a wheelchair ramp to replace stairs for one of the
doorways into the school. The door is 1.8 feet above the ground. To meet the
Americans with Disabilities Act the ramp from the ground to the door must be
at a 2.86o with the ground. To the nearest hundredth of a foot, what is the horizontal
Distance covered by the ramp on the ground?
They are asking for the horizontal
distance on the ground not the length
of the actual ramp itself. Be sure to
carefully read the directions.
Help
Back
Username: mrfahy
Password: geometry
Page 528
Example 5
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