Transcript 1.6 Trigonometric Functions

Trigonometric Review
1.6
Unit Circle
The six trigonometric functions of a right triangle, with an
acute angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are:
hyp
 the side opposite the acute angle ,
opp
 the side adjacent to the acute angle ,
θ
 and the hypotenuse of the right triangle.
adj
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
sin  =
cos  = adj
tan  = opp
hyp
hyp
adj
csc  =
hyp
opp
sec  = hyp
adj
cot  = adj
opp
Calculate the trigonometric functions for  .
5
4

3
The six trig ratios are
4
sin  =
5
4
tan  =
3
5
sec  =
3
3
cos  =
5
3
cot  =
4
5
csc  =
4
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with two sides of
length 1.
45
x 2
1x
(1x) 2  (1x) 2  x 2
45
1x
The Pythagorean Theorem implies that the hypotenuse
is of length 2 .
Geometry of the 30-60-90 triangle
Consider an equilateral triangle with
each side of length 2.
30○ 30○
The three sides are equal, so the
angles are equal; each is 60.
2
The perpendicular bisector
of the base bisects the
opposite angle.
60○
Use the Pythagorean Theorem to
find the length of the altitude, 3 .
2
3
1
60○
2
1
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0
sin x
0

2
1
2
2
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2
1

3
2
2
5
2
x
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0
cos x
1

2
0
2
2
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = cos x
y
3

2



1

2
2
1

3
2
2
5
2
x
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x

x  k  k   
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2

2
 3
2

2
period: 
3
2
x
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot x 
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y  cot x
1. domain : all real x
x  k k  
2. range: (–, +)
3. period: 
4. vertical asymptotes:
x  k k  
vertical asymptotes

3
2
 

2
x  
x0

 3
2
2
x 
x
2
x  2
Graph of the Secant Function
1
sec
x

The graph y = sec x, use the identity
.
cos x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y  sec x
Properties of y = sec x
1. domain : all real x

x  k  (k  )
2
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2
4
y  cos x
x



2
2
4

3
2
2
5
2
3
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x 
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y  csc x
1. domain : all real x
x  k k  
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
x  k k  
where sine is zero.
x



2
2

3 2
2
5
2
y  sin x
4
Graphing
y  a sin( bx  c)  d
a -> amplitude
b -> (2*pi)/b -> period
c/b -> phase shift (horizontal shift)
d -> vertical shift
Angle of Elevation and Angle of Depression
When an observer is looking upward, the angle formed
by a horizontal line and the line of sight is called the:
angle of elevation.
line of sight
object
angle of elevation
horizontal
observer
When an observer is looking downward, the angle formed
by a horizontal line and the line of sight is called the:
horizontal
angle of depression
line of sight
object
observer
angle of depression.
Example 2:
A ship at sea is sighted by an observer at the edge of a cliff
42 m high. The angle of depression to the ship is 16. What
is the distance from the ship to the base of the cliff?
observer
cliff
42 m
horizontal
16○ angle of depression
line of sight
16○
d
42
= 146.47.
tan 16
The ship is 146 m from the base of the cliff.
d=
ship
Example 3:
A house painter plans to use a 16 foot ladder to reach a spot
14 feet up on the side of a house. A warning sticker on the
ladder says it cannot be used safely at more than a 60 angle
of inclination. Does the painter’s plan satisfy the safety
requirements for the use of the ladder?
ladder
house
14
16
sin  =
= 0.875
14
16
θ
Next use the inverse sine function to find .
 = sin1(0.875) = 61.044975
The angle formed by the ladder and the ground is about 61.
The painter’s plan is unsafe!
Fundamental Trigonometric Identities for 0 <  < 90.
Cofunction Identities
sin  = cos(90  )
tan  = cot(90  )
sec  = csc(90  )
cos  = sin(90  )
cot  = tan(90  )
csc  = sec(90  )
Reciprocal Identities
sin  = 1/csc 
cot  = 1/tan 
cos  = 1/sec 
sec  = 1/cos 
tan  = 1/cot 
csc  = 1/sin 
Quotient Identities
tan  = sin  /cos 
cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1
Pg. 51 & 52
tan2  + 1 = sec2 
cot2  + 1 = csc2 
Trig Identities
2 sin   1
2
2 cos   cos   1
2
Homework

READ section 1.6 – IT WILL HELP!!

Pg. 57 # 1 - 75 odd