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22-2 and 22-3: Trig Ratios

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22-2 and 22-3: Trig Ratios
22-2 and 22-3: Trig Ratios
Objectives:
1. To discover and use
the three main
trigonometric ratios
•
•
•
•
Assignment:
P. 311: 12-14
P. 314: 7-9
P. 317-318: 5-10
More on Tangent
Ratios Worksheet
Objective 1
You will discover
and use the three
main
trigonometric
ratios
Warm-Up 1
Find the value of x.
Warm-Up 1
Find the value of x.
History Lesson
Right triangle
trigonometry is the
study of the relationship
between the sides and
angles of right triangles.
These relationships can
be used to make
indirect measurements
like those using similar
triangles.
History Lesson
Early mathematicians
discovered trig by measuring
the ratios of the sides of
different right triangles. They
noticed that when the ratio of
the shorter leg to the longer
leg was close to a specific
number, then the angle
opposite the shorter leg was
close to a specific number.
Example 1
In every right triangle in which the ratio of the
shorter leg to the longer leg is 3/5, the
angle opposite the shorter leg measures
close to 31. What is a good
approximation for x?
Example 2
In every right triangle in which the ratio of the
shorter leg to the longer leg is 9/10, the
angle opposite the shorter leg measures
close to 42. What is a good
approximation for y?
Trig Ratios
The previous examples worked because the
triangles were similar since the angles
were congruent. This means that the
ratios of the sides are equal.
In those cases we were using the tangent
ratio. Here’s a list of the three you’ll have
to know.
sine
cosine
tangent
Investigation 1
Use the Geogebra
Activity to discover
the three main
Trigonometric ratios
sine, cosine, and
tangent.
Summary

A
side adjacent Θ
B
side opposite Θ
C
sin  opposite
hypotenuse
cos  adjacent
hypotenuse
tan  opposite
adjacent
Summary

A
side adjacent Θ
B
side opposite Θ
C
sin  Oh
Hell
cos  Another
Hour
tan 
Of
Algebra
SohCahToa
Soh
sin  opposite
hypotenuse
Cah
cos  adjacent
hypotenuse
Toa
tan  opposite
adjacent
Example 3
Find the values of the six trig ratios for α and
β.
Activity: Trig Table
On the previous example, we knew all the
sides of the triangle, and we just listed the
three trig ratios for those sides using a
generic angle. Usually, though, you know
the angle, and you want to find a side.
Nowadays, we would use a calculator to find
the sine or tangent of an angle. In the
long, dark years before the calculator,
people had to find their trig ratios in a table.
Activity: Trig Table
In the 1500s, Georg Rheticus, a
student of Copernicus, was
the first to define the six trig
functions in terms of right
triangles. He was also the
first to start a book of values
for these ratios, accurate to
ten decimal places to be used
in astronomical calculations.
Activity: Trig Table
Of course, he died before it was
completed, and it was up to his
student, Valentin Otto, to finish
the 1500 page book.
We’re going to do something
similar, but ours will only be
accurate for 3 decimal places,
and probably wouldn’t be too
reliable for astronomical
calculations.
Investigation 2
Use the triangle below to
calculate the sine,
cosine, and tangent of
20.
C
m AC = 15 cm
m CB = 5 cm
20
A
m AB = 14 cm
B
Investigation 2
Use the triangle below to
calculate the sine,
cosine, and tangent of
20.
C
m AC = 14.9 cm
m CB = 5.1 cm
20
A
m AB = 14.0 cm
B
Investigation 2
Use the triangle below to
calculate the sine,
cosine, and tangent of
20.
C
m AC = 14.92 cm
m CB = 5.10 cm
20
A
m AB = 14.02 cm
B
Investigation 2
Use the triangle below to
calculate the sine,
cosine, and tangent of
20.
C
m AC = 14.923 cm
m CB = 5.104 cm
20
A
m AB = 14.023 cm
B
Activity: Trig Table
Step 1: On a sheet of graph paper or
scratch paper, use your protractor to make
as large a right Δ as possible with
∠ = 90°, ∠ = 20°, and ∠ = 70°.
Activity: Trig Table
Step 2: Measure sides , , and  with
your ruler to the nearest millimeter.
Activity: Trig Table
Step 3: Set up a table of values like so:
θ
20°
70°
sin θ
cos θ
tan θ
Activity: Trig Table
Step 4: Now use your calculator to round each calculation
to the nearest thousandths place.
θ
20°
70°
sin θ
cos θ
tan θ
Activity: Trig Table
Step 5: Finally, let’s check your values with
those from the calculator.
For sin, cos, and tan
1. Make sure your calculator is set to DEGREE in the
MODE menu.
2. Use one of the 3 trig keys. Get in the habit of
closing the parenthesis.
Example 4
To the nearest meter,
find the height of a
right triangle if one
acute angle
measures 35° and
the adjacent side
measures 24 m.
Example 5
To the nearest foot, find the length of the
hypotenuse of a right triangle if one of the
acute angles measures 20° and the
opposite side measures 410 feet.
Example 6
Use a special right triangle to find the exact
values of sin(45°) and cos(45°).
Example 7
Find the value of x to the nearest tenth.
1. x =
2. x =
3. x =
22-2 and 22-3: Trig Ratios
Objectives:
1. To discover and use
the three main
trigonometric ratios
•
•
•
•
Assignment:
P. 311: 12-14
P. 314: 7-9
P. 317-318: 5-10
More on Tangent
Ratios Worksheet
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