Right Triangle Trigonometry
Welcome to the wonderful world
of Trigonometry
Trigonometry
Geometry EOCT Domain IV
Trigonometry:
“The study of
Right Triangles”
Trigonometry
Have you ever wondered
how highway engineers are able to make
two sections of a highway overpass meet?
How astronomers find the distances
between planets?
How sailors navigate the high seas?
Trigonometry
They all use trigonometry!
The simple study of the ratios,
sides, and angles of a triangle
is used in surveying,
navigation and in science.
Trigonometry
Trig functions are an
indispensable tool of
science, engineering, and
higher mathematics.
Trigonometry
Trig functions are also used
in descriptions of
vibrations, light, sound,
electrical currents and
planet’s orbital paths.
Basic Trigonometry Rules:
These formulas ONLY work in a right
triangle.
The hypotenuse is across from the right
angle.
Questions usually ask for an answer to the
nearest unit.
You need a scientific or graphing
calculator
Trigonometry
It’s all Greek to me!
You should know these Greek Letters :
Ө
β
α
Trigonometry
Ө This is the symbol for the Greek
letter for T – or “Theta”
You will see Ө, called “Angle Theta”
to represent an Angle T.
Trigonometry
β This is the symbol for the
Greek letter for B – or “Beta”
α This is the symbol for the
Greek letter for A – or “Alpha”
Before we get started….
A Quick Refresher
Parts of a Right Triangle
You must be able to identify the
HYPOTENUSE
The HYPOTENUSE IS the
longest side and it NEVER
touches the right angle.
Picture a child sitting on the floor,
playing marbles…..
you imagine the
Can NowSee
“legs”?
his kid ties a string
around his feet…
This “string”
is the
hypotenuse
As a point of reference,
The triangle we are using comes
from something we call the
“UNIT CIRCLE”.
The Unit Circle is a circle with it’s
center at the origin and a radius
of 1.
The Unit Circle
The unit circle
1
is used to
define trig
ratios for
“ROTATIONS”
You’ll learn more about this in
Algebra II.
Let’s get started…
In every
triangle,
there are 3 angles
and 3 sides.
If it is a right triangle,
there is one right angle
and
two acute
(complementary) angles.
Better yet, name the
angle “Angle Theta”
Name one of the
acute angles “Angle T”
Ө
T
Ө
Remember, it doesn’t matter
which angle you call
“Angle Theta”
Ө
You’ve learned how to use the
Pythagorean Theorem and the
Distance Formula to find the
length of the sides of a triangle.
Another way to find the lengths
of the sides of right triangles is
to use the sine, cosine or
tangent ratios.
The relationships of the 3 sides
of a triangle can be used to
create ratios.
The value of the ratios is
determined by the angles, it
does not depend on the size of
the triangle.
Therefore, the trig
functions work no matter
how big or how small the
triangles are!
There are 6 trigonometric
ratios:
Sine Cosecant
Cosine Secant
Tangent Cotangent
There are 6 trigonometric
ratios.
Formost purposes, 3 ratios
– the Sine, Cosine and
Tangent ratios are enough.
When first learning about the
Trigonometric Ratios, you will be
working in Degree mode.
The calculator will default to
Radian mode, so ....
start by setting the
calculator's MODE to
Degree!!!!
We are going to learn these 3
ratios: Sine, Cosine and Tangent.
Takea moment to find them on
your calculator
COS
Recall from the
“Triangle Inequality Theorem” that there is
a relationship between the angles and the
opposite sides.
Remember a point or a Vertex is
represented by the Capital letter
t
T
Adjacent: Sharing
the same vertex Vocabulary
but having no
interior points in
common. Side
t
T
This leg (side) is adjacent to angle T
Vocabulary This side
is
Opposite: Opposite
from
across from, away from,
angle T
does not touch
Side
t
T
Sketch it out, Label triangles ABC
Copy the parts of this Right
Triangle on your paper
This is
triangle 1 t
1
T
This is
Locate & Label Side T
1. Locate the Right Angle
2. Locate the Hypotenuse.
Side
t
T Ө
This is You could also call it
angle T angle Theta or
Trigonometry Ratios are:
In reference to angle T
Angle T, called “Angle Theta” for the Greek
letter for T
This is the symbol
for Theta: Ө.
Ө
T
Locate the Hypotenuse, mark
it with an “H”
In reference to angle T
Use the 1st letter of each side:
Ө
T
Locate the Adjacent side,
(it touches )
mark it with an “A”
In reference to angle T
Ө
T
A djacent
Locate the Opposite side,
it does not touch
mark it with an “O”
In reference to angle T
O
P
p
o
s
Ө it
T e
A djacent
Triangle 1 should look like this:
H
O
Ө
T
A
Repeat the process for Triangle 2
* Change the location of angle theta:
Ө H
B
Repeat the process for Triangle 2
A, O, and H should be marked.
* Change the location of angle theta:
A Ө H
B
O
The Sine, Cosine, and Tangent
Ratios are:
Copy the formula and ratio for each ratio.
These ratios work regardless of
which acute angle is labeled theta –
but you must work in reference to
it’s location.
The Sine Ratio
This is read “The sine of Angle Theta”
Sin Ө = length of the opposite side
length of hypotenuse
O H O
H
Ө
A
The Cosine Ratios:
Cosine Ө = length of adjacent leg
length of hypotenuse
A
H O
H
Ө
A
The Tangent Ratio is
the only one
The Tangent Ratios: that
doesn’t have the
TangentӨ = length of the opposite side
in it!
Hypotenuseadjacent leg
length of
0
H O
A
Ө
A
The Sine, Cosine, and Tangent
Ratios are:
S oh
C ah
T oa H O
Ө
A
Use the acronyms to make a
beat…
S oh
C ah
T oa H O
Ө
A
Solving Problems
Let’s practice finding the
trigonometric ratios, side
lengths and angle
measurements.
Remember, the steps are the same for
solving problems regardless of which ratio
you are trying to find. Therefore, interchange
TAN, SIN and COS as needed.
Solving Problems
Depending on what information
you are given, and what you are
looking for, there are specific
formulas and different ways to
solve each problem.
Copy and work each problem, make notes so
that you have examples to work by later.
How to solve a problem that
Asks for: the ratio an angle
given two side lengths:
Problem 1
Problem 1:
Write the Tan ratio of angle theta.
Notice that the problem asks for the ratio –
not an angle measurement.
This is simply writing the correct ratio
and simplifying the fraction.
10 6
6
Ө
8 (simplify if possible)
8
Problem 1:
Write the Tan ratio of angle theta.
Solution:
Recall that Tan is “T OA”.
Identify the “Opposite Side”
Identify the “Adjacent Side”
Write the ratio. 10 6
6
Ө
8 (simplify if possible)
8
How to solve a problem that
Asks for: the ratio an angle
given the angle’s degree :
Problem 2
Problem 2:
Find the sine ratio of 35 °
Notice the problem asks for a
“ratio” given and angle
measurement.
You will use the SIN key to solve.
Problem 2:
Find the sine ratio of 35 °
Solution:
In your calculator, press the sin button
A “(“ will appear, enter (35)
Hit enter, the answer is: .5735764364
Round this off unless you are making
further calculations – if you are, keep the
full value and continue working.
Problem 2:
Find the sine ratio of 35 °
Answer:
The answer is: .5735764364
Or
“The sine ratio of a 35° is approximately
.57”
The sine ratio is NOT the angle
measurement
Sometimes, you will use a
special key called the
“Inverse Key”,
The 2nd Key: “2nd Tan”, “2nd Sin”, “2nd Cos”
The ARC key: “Arctan”, “Arcsin”, “Arccos”
Note: The key looks like a negative
exponent but it IS NOT AN EXPONENT
Example:
-1
Tan
Let’s try it!
Locate the inverse key on your
calculator.
How to solve a problem that
Asks for: the ratio an angle
given two side lengths:
Problem 3
Problem 3: A
Find m to the nearest
degree, given cos = 3/5 H
Remember: the COSINE ratio is “CAH” or
the COS = A divided by H
Given, the measurement of the adjacent
angle is 3 units and the measurement of
the hypotenuse is 5 units.
Task: Find m to the nearest
degree, given cos = 3/5
SOLUTION:
Enter 2nd cos (the inverse or arccos)
COS -1 will appear
Input (3/5) and hit enter
the answer is:
53.13010235, or the measurement of
angle is about 53°
How to solve a problem that
asks for the measurement of
an angle
given two side lengths:
Problem 4:
Given:In right triangle ABC,
leg BC=15, and leg AC=20.
the measurement of
Find
angle A to the nearest
degree.
Sketch it out.
A
20
C B
15
In reference to Angle A, or Ө
A What is the
relationship
Ө of 15?
20 It’s opposite
C B
15
In reference to Angle A, or Ө
A What is the
relationship
Ө of 20?
20 It’s Adjacent
C B
15
Which ratio has
opposite and Adjacent?
A TOA
OR
Ө
THE TAN
20 RATIO
C B
15
SET IT UP
TAN (ANGLE A) = 15/20
OR TAN (Ө) = 15/20
Divide 15 by 20 to change the fraction into
a decimal.
The TAN Ratio of (ANGLE A) = 0.75
SET IT UP
Now, to find an angle whose TAN
ratio is 0.75, use your scientific
or graphing calculator.
Use the Inverse button when
finding an angle given two sides
SET IT UP
You now need to activate the Tan-1 key
(it is located above the Tan key).
To activate this Tan-1 key, press 2nd (or
shift) and then the Tan key.
On the graphing calculator, activate the
Tan-1 first, and then enter (0.75.) then
hit enter
Round as directed
Problem 5: find the measurement
of , Given In right triangle ABC,
leg BC=15 and side AB=17.
Find angle A to the nearest degree.
We will be using the same set up and
strategy that was previously
discussed –with one small change.
See if you can find it.
Sketch it out.
A
17
C B
15
Try this: find the measurement of
, round to the nearest degree.
Solution: Opposite and
Hypotenuse are given.
This is a Sine ratio.
Solution: this is a Sine ratio.
Set up the ratio, find an angle whose sine
is 0.75. To do this, use your scientific or
graphing calculator. (On the scientific
calculator, enter 0.75.
Activate the sin-1 key (it is located above
the sin key). On the graphing calculator,
activate the sin -1 first, and then enter
(0.75.)
Problem:
Find the length of a leg
Given the hypotenuse
and an angle
Problem:
Given: In right triangle ABC,
hypotenuse AB=15
and angle A=35º.
Find leg BC to the nearest
tenth.
Sketch it out!
Problem: In right triangle ABC, hypotenuse AB=15 and
angle A=35º. Find leg BC to the nearest tenth.
Draw a picture depicting the
situation.
Be sure to place the degrees
INSIDE the triangle.
A B
Problem: In right triangle ABC, hypotenuse AB=15 and
angle A=35º. Find leg BC to the nearest tenth.
C
?
35°
A B
15
Problem: In right triangle ABC, hypotenuse AB=15 and
angle A=35º. Find leg BC to the nearest tenth.
Place a stick figure at the angle as a point of reference.
Thinking of yourself as the stick figure, label the
opposite side (the side across from you), the
hypotenuse (across from the right angle), and the
adjacent side (the leftover side).
Hypotenuse = 15
Problem: In right triangle ABC, hypotenuse AB=15 and
angle A=35º. Find leg BC to the nearest tenth.
Notice how the items on the sides of the
triangle pair up. The h pairs with the 15,
the o pairs with the x, but the a stands
alone. The a has no companion
term. This means that the a is NOT
involved in the solution of this
problem. Cross it out!
This problem deals with o and h which
means it is using sin A.
Solve for the length of a leg given
hypotenuse and an angle
Answer: set it up then solve algebraically
SinA = opposite leg/hypotenuse
sin35° = x/15
.5736 = x/15
(Multiply both sides by 15)
x = 8.604 ≈ 8.6
Same Problem, different
explanation…
Solution:
Place the degrees in the formula for angle A.
Replace o and h with their companion terms.
Using your scientific/graphing calculator, determine the value of
the left side of the equation. (On most scientific calculators,
press 35 first and then press the sin key. On most graphing
calculators, press the sin key first and the 35 second.)
Solve the equation algebraically. Cross multiply and solve for
x. Or just remember that if the x is on the top, you will multiply to
arrive at your answer. If x is on the bottom, divide to arrive at
your answer (see next example).
Round answer to the desired value.
One more thing…
Angle of Elevation
Angle of Depression
Line of Sight
Angle of Elevation/Angle of Depression:
The angle between a horizontal
line and the line of sight.
If looking down, the angle is an
angle of DEPRESSION. If
looking up, the angle is an
angle of ELEVATION.
The angles between a horizontal
line and the line of sight.
Angle of Elevation If looking up, the
angle is an angle of ELEVATION.
Angle of Depression If looking
down, the angle is an angle of
DEPRESSION.
Line of Sight – The direction of What
you can see
Angle of
Depression.
Line of
Sight
Angle of
Elevation
"You can do this!!!
Trigonometry is easy!"
Now it’s your turn to enter the wonderful world of TRIGONOMETRY!
Begin your assignment:
Make sure you have completed this Graphic
Organizer
Tangent Ratio 13-4, Page 564 Read then do
problems 8 – 20 all.