Objectives:
1. To apply the 6 trig
ratios to right
triangles
2. To use the exact trig
values of special
right triangles
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Assignment:
P. 308: 1-4 S
P. 308: 9-16 S
P. 309: 43-52 S
P. 309: 53-58 S
P. 309: 59-62
P. 309-310: 63-70 S
Homework
Supplement
You will be able to
apply the 6 trig
ratios
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.
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.
(More fancy dress)
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?
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?
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 six you’ll have to know.
sine
cosecant
cosine
secant
tangent
cotangent

A
side adjacent θ
B
side opposite θ
C
sin  opposite
hypotenuse
cos  adjacent
hypotenuse
tan  opposite
adjacent

A
side adjacent θ
B
side opposite θ
C
sin  Oh
Hell
cos  Another
Hour
tan 
Of
Algebra
Soh
sin  opposite
hypotenuse
Cah
cos  adjacent
hypotenuse
Toa
tan  opposite
adjacent
Cho
csc  hypotenuse
opposite
Sha
sec  hypotenuse
adjacent
Cao
cot  adjacent
opposite
Find the values of the six trig ratios for α and β.
On the previous example, we knew all the sides
of the triangle, and we just listed the six 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.
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.
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.
Step 1: On a sheet of graph paper or scratch
paper, use your protractor to make as large a
right ΔABC as possible with m<B = 90°, m<A =
20°, and m<C = 70°.
Step 2: Measure sides AB, AC, and BC with your
ruler to the nearest millimeter.
Step 3: Set up a table of values like so:
θ
20°
70°
sin θ
cos θ
tan θ
csc θ
sec θ
cot θ
Step 4: Now use your calculator to round each calculation to the
nearest thousandths place.
θ
20°
70°
sin θ
cos θ
tan θ
csc θ
sec θ
cot θ
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 Degrees in the Mode
menu.
2. Use one of the 3 trig keys. Get
in the habit of closing the
parenthesis.
For csc, sec, and cot
1. Make sure your calculator is
set to Degrees in the Mode
menu.
2. Type 1/sin(20) for csc 20
3. Type 1/cos(20) for sec 20
4. Type 1/tan(20) for cot 20
Find the value of x to the nearest tenth.
1. x =
2. x =
3. x =
If you are looking up, then the angle from the
horizontal up to the line of sight is the angle
of elevation.
If you are looking down, then the angle from the
horizontal down to the line of sight is the
angle of depression.
A surveyor stands 115 from
the base of the Washington
Monument. She measures
the angle of elevation to
the top of the monument
as 78°18’. How tall is the
Washington Monument?
You will be able to use the exact trig
values of special right triangles
If we take a square and add a diagonal, we
create a 45-45-90 special right triangle.
If the length of the
square’s side is 1, what
is the length of the
hypotenuse?
Now fill in the trig values
for 45° on your table.
If we take an equilateral triangle and add a
height, we create a 30-60-90 right triangle.
If the length of the original
triangle’s side is 2, what
is the length of the
other two sides?
Now fill in the trig values
for 45° on your table.
Find the value of each variable. Write your
answer in simplest radical form.
1.
2.
3.
Find the value of each variable. Write your
answer in simplest radical form.
1.
2.
3.
A 13-meter flagpole bearing the Dragon Banner
casts a 13-meter shadow. Find the angle of
elevation to the sun.
Objectives:
1. To apply the 6 trig
ratios to right
triangles
2. To use the exact trig
values of special
right triangles
•
•
•
•
•
•
•
Assignment:
P. 308: 1-4 S
P. 308: 9-16 S
P. 309: 43-52 S
P. 309: 53-58 S
P. 309: 59-62
P. 309-310: 63-70 S
Homework
Supplement