Download Right Triangle Definitions of Trigonometric Functions

CHAPTER 1.3
CHAPTER 1 TRIGONOMETRY
PART 3 – Right Triangle Trigonometry
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 2.0 – Students know the definition of sine and cosine as y-and x-coordinates of
points on the unit circle and are familiar with the graphs of the sine and
cosine functions.
2
2
 3.0 - Students know the identity cos (x) + sin (x) = 1:
3.1 - Students prove that this identity is equivalent to the Pythagorean
theorem (i.e., students can prove this identity by using the
Pythagorean Theorem and, conversely, they can prove the
Pythagorean Theorem as a consequence of this identity).




3.2 - Students prove other trigonometric identities and simplify others by
2
2
using the identity cos (x) + sin (x) = 1. For example, students use
2
2
this identity to prove that sec (x) = tan (x) + 1.
5.0 – Students know the definitions of the tangent and cotangent functions and
can graph them.
6.0 – Students know the definitions of secant and cosecant functions and
can graph them.
12.0 - Students use trigonometry to determine unknown sides or angles in right
triangles.
19.0 - Students are adept at using trigonometry in a variety of applications
and word problems.
OBJECTIVE(S):
 Students will learn the definition of the six trigonometric functions as
they pertain to the opposite and adjacent legs and hypotenuse of a right
triangle.
 Students will gain further practice in evaluating trigonometric
functions.
 Students will learn the co function, reciprocal, quotient, and
Pythagorean identities.
 Students will gain further practice evaluating trigonometric functions
with a calculator.
 Students will learn how to apply trigonometry to the real-world.
CHAPTER 1.3
The Six Trigonometric Functions
Our second look at the trigonometric functions is from a right triangle perspective.
Consider a right triangle, with one acute angle labeled  . Relative to the angle  , the
three sides of the triangle are the hypotenuse, the opposite side (the side opposite the
angle  ), and the adjacent side (the side adjacent to the angle  ).
Using the lengths of these sides, you can form six ratios that define the six trigonometric
functions of the acute angle  .
sine
cosine
tangent
cosecant
secant
cotangent
In the following definitions it is important to see that 00    900 (  lies in the first
quadrant) and that for such angles the value of each trigonometric function is positive.
Right Triangle Definitions of Trigonometric Functions
Let  be an acute angle of a right triangle. The six trigonometric functions of the angle
 are defined as follows (Note that the functions in the second row are the
_______________________ of the corresponding functions in the first row):
sin  =
opp
hyp
csc =______
cos =
adj
hyp
sec = ______
tan =
opp
adj
cot  = _______
The abbreviations “opp”, “adj”, and “hyp” represent the lengths of the three sides of a
right triangle.
CHAPTER 1.3
opp = the length of the side opposite  .
adj = the length of the side adjacent to  .
hyp = the length of the hypotenuse.
O
SOHCAHTOA
H
A
H
O
A
EXAMPLE 1: Evaluating Trigonometric Functions
Find the exact values of the six trigonometric functions of  .
4
Hypotenuse
3
b gb gbg
By the Pythagorean Theorem, hyp  opp  adj , it follows that
2
hyp
2
=
________________________
=
_____________
=
_________
2
So, the six trigonometric functions of  are
sin  = ______
cos = _______
tan = __________
csc = ______
sec = _______
cot  = __________
CHAPTER 1.3
Often, you will be asked to find the trigonometric functions of a given acute angle  . To
do this, construct a right triangle having  as one of its angles.
EXAMPLE 2: Evaluating Trigonometric Functions of 45 0
Find the values of sin 450 , cos 450 , and tan 450 .
Construct a right triangle having 45 0 as one of its acute angles. Choose the length of the
adjacent side to be ____. From geometry, you know that the other acute angle is also
______. So, the triangle is _________________ and the length of the opposite side is
also _____. Using the Pythagorean Theorem (or your knowledge of special right
triangles), you find the length of the hypotenuse to be ______.
sin 450 
opp

hyp
=
cos 45 0 
adj

hyp
=
tan 45 0 
opp

adj
=
EXAMPLE 3: Evaluating Trigonometric Functions of 300 and 60 0
Use the equilateral triangle shown to find the values of sin 60 0 ,
cos 60 0 , sin 30 0 and cos 30 0 .
1
1
Use the Pythagorean Theorem (or your knowledge of special right triangles) and the
equilateral triangle to verify the lengths of the sides shown in the figure. For  = _____,
you have adj = _____, opp = ______, and hyp = _____. So,
CHAPTER 1.3
sin 60 0 
opp

hyp
and
cos 60 0 
adj

hyp
For  = _____, you have adj = _____, opp = ______, and hyp = _____. So,
sin 30 0 
opp

hyp
and
cos 30 0 
adj

hyp
Sines, Cosines, and Tangents of Special Angles
sin 30 0  sin
sin 45 0  sin
sin 60 0  sin

6

4

3

cos 30 0  cos

cos 45 0  cos

cos 60 0  cos

6

4

3

tan 30 0  tan

tan 45 0  tan

tan 60 0  tan

6

4

3



Note that sin 300  ____ = cos 600 . This occurs because 300 and 60 0 are
________________________ angles. In general, it can be shown from the right triangle
definitions that co functions of complementary are equal. That is, if  is an acute angle,
the following is true.
DAY 1
Relationships with Trigonometry and Complementary Angles
______________.
sin  =
b
What else equals ?
c
c
b
a
Co-function Identities
sin 900   = __________________
c
h
csc 900   = __________________
c
h
sec 900   = __________________
c
h
cot 900   = __________________
cos 900   = __________________
tan 900   = __________________
c
h
c
h
c
h
CHAPTER 1.3
Trigonometric Identities
In trigonometry, a great deal of time is spent studying relationships between
trigonometric functions (identities).
Fundamental Trigonometric Identities
Reciprocal Identities
sin  = ______
cos = _______
tan = __________
csc = ______
sec = _______
cot  = __________
Quotient Identities
tan = ________________
cot  = ____________________
y
Pythagorean Identities
x2  y2  1
or
2
y  x2  1
sin 2 
+
cos 
=
1
sin 2 
+
cos2 
=
1
sin 2 
+
cos2 
=
1
Unit Circle
x
2
CHAPTER 1.3
Note that sin 2  represents __________, cos2  represents ____________, and so on.
EXAMPLE 4: Applying Trigonometric Identities
Let  be an acute angle such that sin   0.6 . Find the values of a.) cos  and b.)
tan  using trigonometric identities.
a.) To find the value of cos  , use the Pythagorean Identity.
sin 2   cos 2   1
Substitute ______ for sin  .
Subtract _______ from each side.
Extract the positive square root.
b.) Now, knowing the sine and cosine of  , you can find the tangent of  to be
tan  
sin 
cos 
1
0.6
sin 2   cos 2   1
cos  = _____
tan  = ______
CHAPTER 1.3
EXAMPLE 5: Applying Trigonometric Identities
Let  be an acute angle such that tan   3 . Find the values of a.) cot  and b.)
sec  using trigonometric identities.
a.) cot  
1
tan 
Reciprocal identity.
b.) sec 2   1  tan 2 
Pythagorean identity.
3
1
sin 2   cos 2   1
cot  = _____
sec  = ______
CHAPTER 1.3
1.) Use the given function values, and trigonometric identities to find the indicated
trigonometric functions.
sec   5 , tan   2 6
a. cos
b. cot 

c. cot 90 0  

d. sin 
2.) Use trigonometric identities to transform one side of the equation into the other.
a. cos sec  1
cos sec  1
Write original equation.
Reciprocal identity.
Divide out common factor.
b
gb
g
b. sec   tan  sec   tan   1
bsec  tan gbsec  tan g 1
Write original equation.
Distributive Property.
Simplify.
Pythagorean Identity.
DAY 2
CHAPTER 1.3
Evaluating Trigonometric Functions with a Calculator
To use a calculator to evaluate trigonometric functions of angles measured in degrees,
first set the calculator to _______________ mode.
Function
a. ) cos 280
Mode
Display
b.) sec 28 0
EXAMPLE 6: Using a Calculator
Use a calculator to evaluate sec 50 40 '12" .


Begin by converting to decimal degree form. [Recall that 1' =______ and 1" = ________].
50 40 '12"
=
=
Then, use a calculator to evaluate ______________.
Function
sec 50 40 '12" =

Display

Applications Involving Right Triangles
Many applications of trigonometry involve a process called solving right triangles. In
this application, you are usually given one side of a right triangle and one of the acute
angles and are asked to find one of the other sides, or you are given two sides and are
asked to find one of the acute angles.
Angle of Elevation – the angle from the horizontal upward to an object.
Object
Angle of
elevation
Observer
Horizontal
CHAPTER 1.3
Angle of Depression – the angle from the horizontal downward to an object.
Horizontal
Observer
Angle of
depression
Object
EXAMPLE 7: Using Trigonometry to Solve a Right Triangle
A surveyor is standing 115 feet from the base of the Washington Monument. The
surveyor measures the angle of elevation to the top of the monument as 78.3 0 . How tall
is the Washington Monument?
You can see that
tan 78.30 =
=
where x = _________ and y is the height of the monument. So, the height of the
Washington Monument is
CHAPTER 1.3
EXAMPLE 8: Using Trigonometry to Solve a Right Triangle
A historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway
to the lighthouse is 400 yards long. Find the acute angle  between the bike path and the
walkway.
You can see that the sine of the angle  is
sin  
=
=
Now you should recognize that  = ______.
By now you are able to recognize that  = ________ is the acute angle that satisfies the
equation sin   ____. Suppose, however, that you were given the equation
sin   ______ and were asked to find the acute angle  . Because
sin 300 
=
and
sin 45 0 

you might guess that  lies somewhere between _____ and ______. In a later section,
you will study a method by which a more precise value of  can be determined.
CHAPTER 1.3
EXAMPLE 9: Solving a Right Triangle
Find the length of c of the skateboard ramp.
c
4 ft
You can see that
=
=
So, the length of the skateboard ramp is
=
=
=

DAY 3