Download Name - White Plains Public Schools

1
Name____________________________
Trigonometric Functions
The Right Triangle
Label the sides of the following right triangles-
Sine =
Cosine =
Tangent =
So….. what is the famous saying to remember this???
R
Find:
T
S
a) sin R
b) cos R
c) tan R
d) sin T
e) cos T
f) tan T
2
Angles As Rotations
Initial Side:
_____________________________________________________________________
_____________________________________________________________________
Terminal Side:
_____________________________________________________________________
_____________________________________________________________________
Standard Position:
_____________________________________________________________________
_____________________________________________________________________
An angle formed by a counter clockwise rotation has a ______________.
An angle formed by a clockwise rotation has a ____________________.
Coterminal Angles:
_____________________________________________________________________
_____________________________________________________________________
If two angles are coterminal, the difference of their measures is 360º or a multiple of 360º.
Notice that 300º - (-60º) = 360º
3
Angles greater than 180°
A wheel makes 5/8 of a complete counter
clockwise rotation <AOA’ =
O
A
A'
A wheel makes ¾ of a complete counter
clockwise rotation.
O
B
<BOB’ = __________
B'
C'
A wheel makes 2/3 of a complete clockwise
rotation.
O
C
<COC’ = ________
Classifying Angles by Quadrant
___< θ < ___
___< θ < ___
___< θ < ___
___< θ < ___
4
Quadrantal Angle:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
ExamplesFind the angle of smallest positive measure that is coterminal with an angle of the given measure.
1) 910°
2) 720°
3) -850°
In which quadrant does an angle of the given measure lie?
4) 200°
5) -40°
Draw an angle of the given measure, indicating the direction of the rotation by an arrow.
6) 130
7) 240
8) -80
9) -740
5
Sine and Cosine as Coordinates
If AB = c, AC = b, BC = a, and m<BCA = 90°,
then:
y
Sin B =
Cos B =
Tan B =
x
Unit Circle- __________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
What would happen if we made c = 1?
sine B =
cos B =
tan B =
Point A has the coordinates (a, b) or (x, y).
So we can conclude that in a unit circle the
sin  =
cos  =
tan  =
Let’s look at each quadrant and the signs of each trigonometric function.
y
y
P
O
y
y
P
R
x
O
R
x
O
R
x
O
P
Quadrant I
Sin  is
Cos  is
Tan  is
Quadrant II
Sin  is
Cos  is
Tan  is
Quadrant III
Sin  is
Cos  is
Tan  is
R
P
Quadrant IV
Sin  is
Cos  is
Tan is
x
6
Examples:
In questions 1 -3 you are given the coordinates of point P, m<ROP = and OR =1.
Find a) sin  b) cos  c) tan 
 1
3
1. P   , 
 2 2 
y
P
O
x
R
y

2
2
2. P   , 
2 
 2
O
R
x
P
y
3. P(.6, -.8)
O
R
x
P
4. If sin  > 0 and cos  <0, what quadrant does an angle of measure  lie in?
5. If sin  < 0 and cos  < 0, what quadrant does an angle of measure  lie in?
7
Now let’s look at the quadrantal angles.
Circle O is a unit circle that intersects the x – axis at A(1,0) and C(-1,0) and the y-axis
at B(0,1) and D (0,-1).
B(0,1)
C(-1,0)
O
A(1,0)
D(0,-1)
Angle of 0°
Angle of 90°
Angle of 180°
Angle of 270°
sin 0° =
sin 90° =
sin 180° =
sin 270° =
cos 0° =
cos 90° =
cos 180° =
cos 270° =
tan 0°
tan 90° =
tan 180° =
tan 270° =
***When is tangent undefined? ___________________________________
8
Function Values of Coterminal Angles
Two angles that are coterminal have the same function values.
Y
R
O
sin 210 ° =
sin -150° =
cos 210° =
cos -150° =
tan 210° =
tan -150° =
X
P
Example: Points A and B are on unit circle O. The coordinates of A are (1,0) and of B

3 1
are   ,  . If m<AOB m = 150, find:
2 2


a) sin 150
b) cos 150
c) tan 150
Function Values of Special Angles
9
Some angles occur frequently in the applications of
trigonometry. We can use some of the relationships
that we learned in the study of geometry to find
exact function values for these angles.
Let <ROP be a central angle
of the unit circle with its
center at the origin. If
m<ROP = 30°, and OP = 1,
then PQ =
3
1
and OQ =
.
2
2
Therefore:
sin 30° = ____
cos 30° = ____
30° and 60°Angles
tan 30° = ____
Equilateral ∆ABC is separated into
two congruent triangles, ∆ACD and
∆BCD, by CD  AB.
C
Therefore,
A
D
B
m<A = ______
m<ACD = ______
m<CDA = ______
If AC = 1, then AD = _________
C
Using the Pythagorean Theorem we can find the length of CD.
30
1
(CD)2 + (AD)2 = (AC)2
60
A
Sin 30 =
Sin 60 =
Cos 30 =
Cos 60 =
Tan 30 =
Tan 60 =
1/2
D
10
P
O
Q
R
Again, let <ROP be a central
angle of the unit circle with its
center at the origin. If m<ROP
= 60°, and OP = 1, then
P
PQ =
O
Q
3
1
and OQ = .
2
2
R
Therefore:
sin 60° = ____
cos 60° = ____
tan 60° = ____
11
o
45 Angles
In the diagram, ∆ABC is an
isosceles right triangle with a
right angle at C. Therefore, m<A
= m<B = 45°, and AC = BC. If
AB = 1, let AC = x and BC = x.
Using the Pythagorean Theorem:
B
1
x
A
C
x
(AC)2 + (BC)2 = (AB)2
Let <ROP be a central angle of
the unit circle with its center at
the origin. If <ROP = 45 and OP
P
O
Q
= 1, Then PQ =
OQ =
R
2
and
2
2
.
2
Therefore:
sin 45° =_____
cos 45° = _____
tan 45° = _____
Summary

sin 
cos 
tan 
0°
30°
45°
60°
90°
12
Examples1) Find the exact numerical value of (sin 60°)(cos 30°) + tan 60°.
2) Find the exact numerical value of (sin 90°) - (cos 60° + cos 30°).
3) You need to fill in the chart for special angles.
******* try to do this with out looking at your table **********

0°
30°
45°
sin 
cos 
tan 
Finding Reference Angles
60°
90°
13
We have learned values for angles whose measures are between 0° and 90° we now
will learn how to relate function values of angles in quadrants II, III, and IV to values
of trigonometric functions of angles in quadrant I.
Reference Angles
Reference Angle_______________________________________________________________
_____________________________________________________________________________

Quadrant I
180 - 
Quadrant II
y
y
O
(-a,b) P
(-cos  ,sin  )

P (a,b)
(cos  , sin  )
R

R
O
x
x
Sin  = sin (180 - )
Cos  = -cos (180 -)
Tan  = -tan (180 – )
Example:  = 150 ; The reference angle is ____.
Sin 150 =
Cos 150 =
Tan 150 =
 - 180
Quadrant III
y
y

O
360 - 
Quadrant IV
R

O
x
R
x
P (a,-b)
(cos  , -sin  )
P (-a,-b)
(-cos  , -sin  )
Sin  = -sin (180 +)
Cos  = -cos (180 +)
Tan  = tan (180 + )
Sin  = -sin (360 - )
Cos  = cos (360 - )
Tan  = -tan (360 - )
Example:  = 240. The reference angle is ____.
Sin 240 =
Example:  = 320. The reference angle is ____.
Sin 320 =
Cos 240 =
Cos 320 =
Tan 240 =
Tan 320 =
14
Summary
If  is the measure of an angle greater than 90° but less than 360°:
90°<  < 180°
180° <  < 270°
270° <  < 360°
Quadrant II
Quadrant III
Quadrant IV
Sin  = sin (180° - )
Sin  = -sin (-180°)
Sin  = -sin (360° - )
Cos  = -cos (180° - )
Cos  = -cos (-180°)
Tan  = -tan (180° - )
Tan  = tan ( – 180°)
Cos  = cos (360° - )
Tan  = -tan (360° - )
ExamplesExpress the given function as a function of a positive acute angle.
1) cos 155°
2) sin 340°
3) tan 200°
4) tan 215°
a)Express the given function as a function of a positive acute angle.
b) Find the exact function value.
5) cos 300°
6) sin 240°
7) cos 405°
8) tan 200°
11) tan 600°
12) sin (-45°)
Find the exact function value.
9) sin 135°
10) sin 240°
Find the exact value of the given expression.
13) cos 135° + cos 225°
14) sin 300° + sin ( -240°)
15
Radian Measure
Radian________________________________________________________________
_____________________________________________________________________
Q
r
O
<POQ = 1 radian
P
r
***Remember C = 2r so 360 = 2r.
Solve for r: r 
360
2
360o = 2 radians
180o =  radians
One radian = 57.3o
In order to convert from degrees
In order to convert from radians to
to radians, multiply by
degrees, multiply by

180
.
Example: Convert 45o to radian
measure.
Example: Convert 90o to radian
measure.
180

Example: Find the degree measure of

an angle of
radians.
4
Example: Find the degree measure of
3
angle of
radians.
2
16
17
Find the radian measure of an angle given the degree measure:
1) 30°
2)120o
3) 100°
4) 300°
Find the degree measure of each angle given the radian measure:
5)

6)
3
5
6
7)

9
*******Convert to radians and MEMORIZE the following********
8) 0°
9) 90°
10) 180°
11) 270°
12) 360°
Trigonometric Functions involving Radian Measure
Since angle measure can be expressed in radians as well as in degrees, we can find
values of trigonometric functions of angles expressed in radian measure.
To do this, we convert the radian measure to a degree measure and follow the
procedures learned earlier.
4
.
3
4 180 720


 240
3

3
Example: Find the exact value of sin
Step 1: Change to degrees:
Step 2: Draw a unit circle with an angle of 240o.
Step 3: Find the reference angle:
18
60
240 - 180 = 60
Step 4 : Find the exact value of the function of the reference
angle.
sin 240 = -sin 60 = -
3
2
ExamplesFind the exact value of each of the following:
1) cos
2
3
2) tan
3
4
3) sin
5
2
4) sin

3
5) cos
4
3
19
6) sin 
7) tan 

6

3
8. If a function f is defined as f(x) = cos 2x + sin x, find the numerical value of

2
f( ).
9. Find the numerical value of f(x) = 3cosx – sin2x of f() for the given
function f.

10.Find the numerical value of f(x) = 2sinx + 2cosx of f   for the given
3
function f.
20
Measure of an angle in Radians s = rθ
To find the measure of an angle in radians, when you are given the length of the arc and radius,
follow the formula below:
Measure of an angle in radians = length of the intercepted arc
length of radius
In general, if θ is the measure of a
central angle in radians, s is the length
of the intercepted arc, and r is the length
of a radius, then:
θ=
s
r
If both members of this equation are
multiplied by r, the rule is stated
s=θr
Examples:
1) In a circle, the length of a radius is 4cm. Find the length of an arc intercepted by a central angle
whose measure is 1.5 radians.
2) Find the radius of a circle if θ = 6 and s = 12.
3) Find the length of the arc when θ = 4 and the radius of the circle is 1.25.
21
4) A circle has radius of 1.7 inches. Find the length of an arc intercepted by a central angle of 2
radians.
5) On a clock, the length of a pendulum is 30 centimeters. A swing of the pendulum determines an
angle of 0.8 radian. Find, in centimeters, the distance traveled by the tip of the pendulum during this
swing.
6) A ball rolls in a circular path that has a radius of 5 inches, as shown in the accompanying diagram.
If the ball rolls through an angle of 2 radians, find the distance traveled by the ball.
7) A ball is rolling in a circular path that has a radius of 10 inches, as shown in the accompanying
diagram. What distance has the ball rolled when the subtended arc is 54°? Express your answer to the
nearest hundredth of an inch.
22
8) The Vietnam Veterans Memorial in Washington, D.C., is made up of two walls, each 246.75 feet
long, that meet at an angle of 125.2˚. Find, to the nearest foot, the distance between the ends of the
walls that do not meet.
9) As shown in the accompanying diagram, a dial in the shape of a semicircle has a radius of 4
centimeters. Find the measure of θ, in radians, when the pointer rotates to form an arc whose length is
1.38 centimeters.
10) The accompanying diagram shows the path of a cart traveling on a circular track of radius 2.40
meters. The cart starts at point A and stops at point B, moving in a counterclockwise direction. What
is the length of minor arc AB, over which the cart traveled, to the nearest tenth of a meter?