Download Right Triangle Trigonometry

RIANGLE
Let's review a few things about inverse functions.
• To have an inverse function, a function must be one-to-one
(remember if a horizontal line intersections the graph of a
function in more than one place it is NOT one-to-one).
•If we have points on a function graph and we trade x and y
places we'll have points on the inverse function graph.
•Functions and their inverses "undo" each other so f
f 1  x
•Since x and y trade places, the domain of the function is the
range of the inverse and the range of the function is the
domain of the inverse
•The graph of a function and its inverse are reflections about
the line y = x (a 45° line).
Is y = sin x a one-to-one function?
1
Consider sin  
2
Is there more than one  that satisfies this
equation?
  30 and   150 are both solutions
as well as any coterminal angle to these angles.
Since we want a one-to-one function that will have
an inverse we are going to restrict  to acute angles.
The following tables list the angles you should know the exact
values of. Remember you can get the first table by drawing 30-6090 or 45-45-90 triangles and the second table contains quadrantal
angles
Find the angle  that satisfies each equation
where 0    90
2
sin  
2
2
Since sin 45 
,   45
2
3
cos  
2
3
Since cos 30 
,   30
2
tan   1
Since tan 45  1,   45
When we are given a function whose input is the
trig function of an angle and the output is the angle,
it is called an inverse trig function. and is denoted
with the inverse function notation. For example:
1  1 
sin    
30
2
cos1 1  
0
tan 1

 3  60
This is called the inverse sine function and
means “what angle  has a sine value of
1/2?” (we’ll look for an acute angle)
This is called the inverse cosine function
and means “what angle  has a cosine
value of 1?” (we’ll look for an acute angle)
This is called the inverse tangent function
and means “what angle  has a tangent
value of 3 ?” (we’ll look for an acute angle)
Definition: Inverse Sine, Cosine
and Tangent Functions
The reason there is a restriction on the range
here is so that this will be a one-to-one function
For help on using your calculator to
compute inverse trig functions, click
here.
In a right triangle, the shorter sides are called legs and the longest side
(which is the one opposite the right angle) is called the hypotenuse
We’ll label them a, b, and c and the angles 
and . Trigonometric functions are defined by
taking the ratios of sides of a right triangle.

adjacent
c
First let’s look at the three basic functions.
leg
b
SINE

leg a
COSINE
TANGENT
They are abbreviated using their first 3 letters
opposite
a
opposite a
sin  

tan  

hypotenuse c
adjacent b
adjacent
b
cos  

hypotenuse c
We could ask for the trig functions of the angle  by using the definitions.
You MUST get them memorized. Here is a
mnemonic to help you.

c
The old Indian word:
b
SOHCAHTOA

adjacent
a
opposite
b
sin  

hypotenuse c
adjacent
a
cos  

hypotenuse c
opposite b
tan  

adjacent a
It is important to note WHICH angle you are talking
about when you find the value of the trig function.

c
5
4b
Let's try finding some trig functions
with some numbers. Remember that
sides of a right triangle follow the
Pythagorean Theorem so
a b c
2

2
2
adjacent
a3
sin  =
o 3

h 5
Let's choose:
tan  =
o 4

a 3
32  4 2  5 2
Use a mnemonic and
figure out which sides
of the triangle you
need for tangent.
sine.
You need to pay attention to which angle you want the trig function
of so you know which side is opposite that angle and which side is
adjacent to it. The hypotenuse will always be the longest side and
will always be opposite the right angle.

Oh,
I'm
acute!
This method only applies if you have
a right triangle and is only for the
acute angles (angles less than 90°)
in the triangle.
5
4

3
So
am I!
There are three more trig functions. They are called the
reciprocal functions because they are reciprocals of the first
three functions.
Oh yeah, this
Like the first three trig functions, these are referred
to by the first three letters except for cosecant since
it's first three letters are the same as for cosine.
opposite
sin 
hypotenuse
means to flip the
fraction over.
hypotenuse
cosecant 
opposite
adjacent
cos 
hypotenuse
hypotenuse
secant 
adjacent
opposite
tan 
adjacent
adjacent
cotangent 
opposite
Best way to remember these is learn which is reciprocal of which and flip them.
5
sec  =
4
Let's try one:

a
h
c
5
Which trig function is this the reciprocal of?
adjacent
hypotenuse
so sec is
cos is
hypotenuse
adjacent
4b
o
cot  =
a
a3

3
4
opposite
adjacent
tan is
so cot is
adjacent
opposite
As a way to help keep them straight I think, The "s" doesn't go with "s" and
the "c" doesn't go with "c" so if we want secant, it won't be the one that
starts with an "s" so it must be the reciprocal of cosine. (have to just
remember that tangent & cotangent go together but this will help you with
sine and cosine).
TRIGONMETRIC IDENTITIES
Trig identities are equations that are true for all angles in the domain. We'll be
learning lots of them and use them to help us solve trig equations.
RECIPROCAL IDENTITIES
These are based on what we just learned.
1
csc  
sin 
1
sec  
cos 
1
cot  
tan 
We can discover the quotient identities if we take quotients of sin and cos:
0
o
sin  h o  h
  tan 
 
cos  a h a
a
h
Which trig function is this?
Remember to simplify
complex fractions you invert
and multiply (take the bottom
fraction and "flip" it over and
multiply to the top fraction).
Try this same thing with
what do you get?
cos 
sin 
and
We can discover the quotient identities if we take quotients of sin and cos:
0
o
sin  h o  h
  tan 
 
cos  a h a
a
h
Which trig function is this?
Remember to simplify
complex fractions you invert
and multiply (take the bottom
fraction and "flip" it over and
multiply to the top fraction).
Try this same thing with
what do you get?
QUOTIENT IDENTITIES
These are based on what we just learned.
sin 
tan  
cos 
cos 
cot  
sin 
cos 
sin 
and
If the angle  is acute (less than 90°) and you have
the value of one of the six trigonometry functions,
you can find the other five.
Reciprocal of sine so "flip" sine over
Sine is the ratio of which
sides of a right triangle?
o
1
sin   
3
h
csc  3
a 1  3
2

1
2
When you know 2 sides of a right
triangle you can always find the 3rd
with the Pythagorean theorem.
a 82 2
3
2 a2
2
a 2 2
cos  
3
h
Draw a right triangle and
label  and the sides you
know.
o
1
tan   
a 2 2
Now find the other
trig functions
sec  3
2 2
cot  2
"flipped"
cos
"flipped"
2 tan
RIGH
APPLICATIONS
To "solve" a right triangle means to find the measurements
all of the angles and lengths of all of the sides. We typically
round sides to two decimal places and angles to one.
If we are given at least one of the sides and one of the
angles (other than the right angle) or another side, we can
solve a right triangle.

c
b

a
We typically label the acute angles  and  and sides
opposite them a and b with the side opposite the right
angle (the hypotenuse) c.
Solve a right triangle if b = 4 and  = 10°
Draw a picture.
What trig function can we use to relate the angle given, the
side given and a side we want to know?
o
sin  
h
4
sin 10 
c
80

b
44
4
c
 23.04
sin 10
Solve for c
Use a calculator
23.04
c
and make sure it is
in degrees.
10

22.69
a
The sum of the angles in a triangle is 180° so you know 
must be 80°.
Use the Pythagorean Theorem to find a.
4  a  23.04
2
2
2
Solve a right triangle if b = 4 and  = 10°
Could I have used tan instead of sin in the last problem?
o
tan  
a
4
tan 10 
a
80

4
a
 22.69
tan 10
Solve for a
23.04
c
44
b
10

Yes---either way
would work.
22.69
a
The sum of the angles in a triangle is 180° so you know 
must be 80°.
Use the Pythagorean Theorem to find c.
4  22.69  c
2
2
2
Solve a right triangle if b = 4 and c = 6
What trig function can we use to relate the two sides given,
and an angle we want to know?
4
sin  
6
o
sin  
h

48.2
  41.8
c6
44
b
Find sin-1(4/6)
Use a calculator
and make sure it is
in degrees.
41.8

4.47
a
The sum of the angles in a triangle is 180° so you know 
must be 180°- 90° - 41.8°.
Use the Pythagorean Theorem to find a.
4 a 6
2
2
2
There are many applications of right triangle trigonometry.
A safety regulation states that the
maximum angle of elevation for a
rescue ladder is 72°. If a fire
department's longest ladder is 110
feet, what is the maximum safe
rescue height?
What trig function relates the
angle we know, the side we know
and the side we want to know?
0
sin  
h
a
sin 72 
110
a  110 sin 72
 104.62 ft
The angle of elevation is measured
from a horizontal line looking up at
something.
The Angle of Depression is measured
from a horizontal line looking down at
something.
An outdoor basketball backboard casts a shadow 17 1/3
feet long. The angle of elevation from a point at the end
of the shadow to the top of the backboard is 35.8°. Find
the height of the backboard.
What would you want to do first?
h
35.8°
17 1/3 ft
shadow
Draw a picture!
Now you see that this is nothing
but a right triangle trig problem.
What trig function would you
use that relates the angle and
side you know to the side you
want to know?
tan 35.8 
h
1
17
3
1
17 tan 35.8  h
3
h  12.5 ft