Chapter 17
17-1 Trigonometric Functions in
Triangles
CONCEPT
Special Right Triangles
SUMMARY
2x
x
x 3
30-60-90 Triangle
The hypotenuse is 2
times the short leg!
The long leg is the 3
times the short leg
x 2
45-45-90 Triangle
The hypotenuse is the
2 times the leg!
x 2
x 2
x  2 5
x 5 2
hyp
leg 
2
x 8 2
21 2
x
2
2x
x
x 3
2x
x
x 3
10 is the short leg since it is
opposite the 30° Angle
Hypotenuse = 2(Short Leg)
y = 2(10)
y = 20
Long Leg  3  Short Leg
x  10 3
CONCEPT
Exact Trig Values
SUMMARY
2x
x
x 2
x 3
1
sin 30 
2
3
cos30 
2
3
tan 30 
3
3
sin 60 
2
1
cos 60 
2
cos 45 
tan 60  3
tan 45  1
2
sin 45 
2
2
2
Find the value of each variable
Find x first
x
tan 72 
3
Find y next
3
cos 72 
y
3 tan 72  x
y(cos 72)  3
x  9.2331
3
y
 9.708
cos 72
Exact
Value
Approximation
Find the value of each variable
Find x first
8
cos 40 
x
Find y next
x(cos 40)  8
y  8(tan 40)  6.7128
8
 10.4433
x
cos 40
Exact
Value
Approximation
tan 40 
y
8
Find the value of each variable
B
A
x  10(tan 25)  4.6631
10
y
 11.0338
cos 25
x = 12(cos 63) = 5.4479
y = 12(sin 63) = 10.6921
SHORT-RESPONSE TEST ITEM
A wheelchair ramp is 3 meters long and inclines at
Find the height of the ramp to the nearest tenth
centimeter.
Y
W
Multiply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
Method 2
The horizontal line from the top of the platform to which
the wheelchair ramp extends and the segment from the
ground to the platform are perpendicular. So,
and
are complementary angles. Therefore,
Y
W
Multiply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
SHORT-RESPONSE TEST ITEM
A roller coaster car is at one of its highest points. It
drops at a
angle for 320 feet. How high was the
roller coaster car to the nearest foot before it began
its fall?
Answer: The roller coaster car was about 285 feet above
the ground.
Evaluate the six trigonometric functions of the angle 
shown in the right triangle.
13
SOLUTION
The sides opposite and adjacent to the angle
are given. To find the length of the hypotenuse,
use the Pythagorean Theorem.
If cos 
8
, find the other 5 trig functions.
17
15
SOLUTION
Draw a triangle such that one angle has the
given cosine value use the Pythagorean
theorem
17 2  82  225  15
15

17
8

17
17

15
17

8
17
8

15
8
8

15
HW #17-1
Pg 732 1-23
Chapter 17
17-2 More fun with Trigonometric
Functions
ANGLES IN STANDARD POSITION
GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS
(x, y)
0
r
Let 0 be an angle in standard position and (x, y) be any
point (except the origin) on the terminal side of 0 . The
six trigonometric functions of 0 are defined as follows.
GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS
y
csc
sin 00 == r r , y  0
y
yr
yr
0
GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS
sec 0 ==
cos
xr
,x0
rx
xr
xr
0
GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS
yx
tan
cot 00==x , x, y 0 0
y
x
y
y
x
0
GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS
(x, y)
0
r
Pythagorean theorem gives
r = x 2 +y 2.
Let  be an angle in standard position and let P(x, y) be a point on the
terminal side of  . Using the Pythagorean Theorem, the distance r
from the origin to P is given by r  x 2  y 2 . The trigonometric
functions of an angle in standard position may be defined as follows.
Evaluating Trigonometric Functions Given a Point
Let (3, – 4) be a point on the
terminal side of an 0 angle in
standard position. Evaluate the
six trigonometric functions
of 0 .
r
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = x2 + y2
= 3 2 + (– 4) 2
=
25
= 5
0
(3, – 4)
Evaluating Trigonometric Functions Given a Point
Using x = 3, y = – 4, and r = 5,
you can write the following:
r
0
(3, – 4)
y
sin 0 = r = – 4
5
x
cos 0 = r =
3
5
y
tan 0 = x = – 4
3
csc 0 = yr = – 5
4
r
sec 0 = x =
5
3
x
cot 0 = y = – 3
4
The values of trigonometric functions of angles greater than
90° (or less than 0°) can be found using corresponding acute
angles called reference angles.
Let 0 be an angle in standard position. Its reference angle
is the acute angle 0' (read theta prime) formed by the terminal
side of 0 and the x-axis.
90 < 0 < 180;

2
< 0 < 
0'
Degrees:
0' = 180 – 0
Radians:
0' =
 –0
0
180 < 0 < 270;
 < 0 <
3
2
0
0'
Degrees:
0' = 0 – 180
Radians:
0' = 0 – 
270 < 0 < 360;
3
2
< 0 < 2
0
0'
Degrees:
0' = 360 – 0
Radians:
0' = 2 – 0
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Use these steps to evaluate a trigonometric function of
any angle 0.
1
Find the reference angle 0'.
2
Evaluate the trigonometric function for angle 0'.
3
Use the quadrant in which 0 lies to determine the
sign of the trigonometric function value of 0 .
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Signs of Function Values
Quadrant II
Quadrant I
sin 0 , csc 0 : +
sin 0 , csc 0 : +
cos 0 , sec 0 : –
cos 0 , sec 0 : +
tan 0 , cot 0 : –
tan 0 , cot 0 : +
Quadrant III
sin 0 , csc 0 : –
Quadrant IV
sin 0 , csc 0 : –
cos 0 , sec 0 : –
cos 0 , sec 0 : +
tan 0 , cot 0 : +
tan 0 , cot 0 : –
Using Reference Angles to Evaluate Trigonometric Functions
Evaluate tan (– 210).
0' = 30
SOLUTION
0 = – 210
The angle – 210 is coterminal with 150°.
The reference angle is 0' = 180 – 150 = 30.
The tangent function is negative in Quadrant II,
so you can write:
tan (– 210) = – tan 30 = –
3
3
 2 2
,


2
2



2 2

,


2
2


2
2
2

2

2
2
,


2
2


1
1

2
2
2
2

2
2
2
2
1
1
2
2

2
2
 2
2
,



2
2



2 2

,


2
2


 3 1
 , 
 2 2
1
1
3
2

2
3
1


21
2

3 1

, 

2
 2

2
2
,


2
2


1
3
2
3
12
 2 2
,


2
2


 3 1
 , 
 2 2
1
2
1
2
 3 1
 , 
2
 2
 2
2
,



2
2



 1 3
 , 
 2 2 

2 2

,


2
2


 3 1
 , 
3
 2 2
2


3
2

1
1
1
2
1

2
1
2
1
2


1
3 1

,



2
2



2
2
,


2
2

 1
3
 ,

2
2


1 3
 , 
 2 2  2 2 
,


2
2


 3 1
3
 , 
2 2

2
1



3
2
1 3
 , 
2 2 
 3 1
 , 
2
 2
 2
2
,



2
2



2 2

,


2
2


 3 1
 , 
 2 2
 1 3
 , 
 2 2 

1 3
 , 
 2 2  2 2 
,


2
2


 3 1
60
 , 
 2 2
45
30


3 1

, 

2
 2

2
2
,


2
2

 1
3
 ,

2
2




1
3
 , 
2 
2
 3 1
 , 
2
 2
 2
2
,



2
2


undef
 3
3
1

1
3
3
3
3
0
0
3
3

1
1
3
 ,

2

2


3
undef
1
 3
3
3
HW #17-2
Pg 739-740 1-61 Odd, 62-63
Chapter 17
17-3 Radians, Cofunctions, and
Problem solving
Theorem 17-1
The radian measure  of a rotation is the ratio of the distance s
traveled by a point at a radius r from the center of rotation to
the length of the radius.
s

r
 1 3
 , 
 2 2 

2 2
 ,

2
2


 3 1
 , 
 2 2
 1,0 
 0,1

1 3
 , 
2 2 
 2 2
,


2
2


 3 1
 , 
 2 2


3 1

, 

2
 2

2
2
,


2
2


 1
3
 ,

2
2



1,0
 3 1
 , 
2
 2

 0, 1
 2
2
,


2
2


1 3
 , 
2 2 
One Radian is the measure of an angle in standard
position whose terminal side intercepts an arc of length r.
The arc length s of a sector with radius r and central angle  is
given by the formula: s = r 
A
B
Linear Speed
Angular Speed
Linear Speed
A child is spinning a rock at the end of a 2-foot rope at
the rate of 180 revolutions per minute (rpm). Find the
linear speed of the rock when it is released.
The rock is moving around a circle of radius r = 2 feet.
The angular speed of the rock in radians is:
The linear speed of the rock is:
The linear speed of the rock when it is released is 2262 ft/min
 25.7 mi/hr.
Linear Speed on Earth Earth rotates on an axis through its
poles. The distance from the axis to a location on Earth 40°
north latitude is about 3033.5 miles. Therefore, a location on
Earth at 40° north latitude is spinning on a circle of radius
3033.5 miles. Compute the linear speed on the surface of Earth
at 40° north latitude.

v  rw  3033.5    794 mph
 12 
40
s
9
  .08 radians  4.58
Co-Function Identities
A  90  B
A  B  90
b
sin B 
c
b
cos A 
c
sin B  cos A
sin B  cos(90  B)
The sine of an angle is the cosine of the complement and the
cosine of an angle is the sine of the complement
The same is true of each trig function and its co-function
HW #17-3
Pg 746-747 1-47 Odd, 48-49
Chapter 17
17-4 Finding Function Values
1
3
 ,

2
2 

 3
undef
3
1
Let's label
the unit
circle with
values of
the tangent.
(Remember
this is just
y/x)

1
3
3
3
3
0
0

3
3
1
1
1
3
 ,

2
2 

3
undef
 3
3
3
What is the measure of this angle?
You could measure in the positive
 = - 360° + 45°
direction and go around another rotation
which would be another 360°
 = - 315°
 = 45°
You could measure in the positive
direction
 = 360° + 45° = 405°
You could measure in the negative
direction
There are many ways to express the given angle.
Whichever way you express it, it is still a Quadrant I
angle since the terminal side is in Quadrant I.
If the angle is not exactly to the next degree it can be
expressed as a decimal (most common in math) or in
degrees, minutes and seconds (common in surveying
and some navigation).
1 degree = 60 minutes
1 minute = 60 seconds
 = 25°48'30"
degrees
seconds
minutes
To convert to decimal form use conversion fractions.
These are fractions where the numerator = denominator
but two different units. Put unit on top you want to
convert to and put unit on bottom you want to get rid of.
Let's convert the
seconds to
minutes
30"  1'
60"
= 0.5'
1 degree = 60 minutes
1 minute = 60 seconds
 = 25°48'30" = 25°48.5' = 25.808°
Now let's use another conversion fraction to get rid of
minutes.
48.5'  1
60'
= .808°
(cos ,sin )

(1,0)

What is the length
of this segment?
HW #17-4
Pg 753 1-73 Odd, 74-78
Chapter 17
17-6 Trig Functions and
Relationships
 1 3
 , 
 2 2 

2 2
 ,

2
2


 3 1
 , 
 2 2
 1,0 
 0,1

1 3
 , 
2 2 
 2 2
,


2
2


 3 1
 , 
 2 2


3 1

, 

2
 2

2
2
,


2
2


 1
3
 ,

2
2



1,0
 3 1
 , 
2
 2

 0, 1
 2
2
,


2
2


1
3
 ,

2
2


(cos ,sin )

(1,0)

Let’s consider the
length of this
segment?
HW #17-6
Pg 767-768 1-33 Odd, 34-37
Chapter 17
17-8 Algebraic Manipulations
Row 2, 4, 6
Row 1. 3. 5
Row 2, 4, 6
Row 1. 3. 5
Row 1, 2, 3. 4, 5, 6
HW #17-8
Pg 772-773 1-47 Odd, 48-51
Test Review
Determine the quadrant in which the terminal side of the
angle lies
Find one positive angle and one negative angle coterminal
with the given angle.
Rewrite each degree measure in radians and each radian
measure in degrees.
Find the arc length of a sector with the given radius r and
central angle 
Evaluate the trigonometric function without using a
calculator.
Evaluate the trigonometric function without using a calculator.
Find the values of the other five trigonometric functions of .
Verify the identity.
Verify the identity.
HW #R-17
Pg 776-778 1-46