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Trigonometric Review. 1.6. Unit Circle. θ. adj. opp. sin = cos = tan = csc = sec = cot =. hyp. adj. hyp. hyp. adj. opp. adj. opp. - PowerPoint PPT Presentation
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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:
the side opposite the acute angle ,
the side adjacent to the acute angle , and the hypotenuse of the right triangle.
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot =
opphyp
adj
hyp
hypadj
adj
opp
oppadj
hyp
opp
Calculate the trigonometric functions for .
The six trig ratios are
4
3
5
sin =5
4
tan =3
4
sec =3
5
cos =5
3
cot =4
3
csc =4
5
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with two sides of length 1.
1x
1x
45
452x
2)1()1( 22 xxx
The Pythagorean Theorem implies that the hypotenuse
is of length .2
60○ 60○
Consider an equilateral triangle with each side of length 2.
The perpendicular bisector of the base bisects the opposite angle.
The three sides are equal, so the angles are equal; each is 60.
Geometry of the 30-60-90 triangle
2 2
21 1
30○ 30○
3
Use the Pythagorean Theorem to find the length of the altitude, . 3
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.
0-1010sin x
0x2
2
32
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
2
3
2
22
32
2
5
1
1
x
y = sin 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.
10-101cos x
0x2
2
32
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
2
3
2
22
32
2
5
1
1
x
y = cos x
y
x
2
3
2
32
2
Graph of the Tangent Function
2. range: (–, +)
3. period:
4. vertical asymptotes: kkx
2
1. domain : all real x kkx
2
Properties of y = tan x
period:
To graph y = tan x, use the identity .x
xx
cos
sintan
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.
Graph of the Cotangent Function
2. range: (–, +)
3. period: 4. vertical asymptotes:
kkx
1. domain : all real x kkx
Properties of y = cot x
y
x
2
2
2
32
3
2
xy cot
0xvertical asymptotes xx 2x
To graph y = cot x, use the identity .x
xx
sin
coscot
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.
2
3
y
x
2
2
2 3
2
5
4
4
xy cos
Graph of the Secant Function
2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:
kkx 2
1. domain : all real x)(
2 kkx
cos
1sec
xx The graph y = sec x, use the identity .
Properties of y = sec x
xy sec
At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.
2
3
x
2
2
2
2
5
y
4
4
Graph of the Cosecant Function
2. range: (–,–1] [1, +) 3. period:
where sine is zero.
4. vertical asymptotes: kkx
1. domain : all real x kkx
sin
1csc
xx To graph y = csc x, use the identity .
Properties of y = csc x xy csc
xy sin
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
Graphing
dcbxay )sin(
a -> amplitude
b -> (2*pi)/b -> period
c/b -> phase shift (horizontal shift)
d -> vertical shift
angle of elevation
When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the:
Angle of Elevation and Angle of Depression
When an observer is looking upward,
angle of elevation.
the angle formed by a horizontal line and the line of sight is called the:
observerobjectline of sight
horizontal
observer
objectline of sight
horizontal
angle of depressionangle 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?
The ship is 146 m from the base of the cliff.
line of sight
angle of depressionhorizontalobserver
ship
cliff42 m
16○
16○
d
d = = 146.47. 16tan
42
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 angleof inclination. Does the painter’s plan satisfy the safetyrequirements for the use of the ladder?
Next use the inverse sine function to find .
= sin1(0.875) = 61.044975
The painter’s plan is unsafe!
ladderhouse1614
The angle formed by the ladder and the ground is about 61.
θsin = = 0.875
16
14
Fundamental Trigonometric Identities for 0 < < 90.Cofunction Identities
sin = cos(90 ) cos = sin(90 )tan = cot(90 ) cot = tan(90 )sec = csc(90 ) csc = sec(90 )
Reciprocal Identities
sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin
Quotient Identities
tan = sin /cos cot = cos /sin
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2 Pg. 51 & 52
Trig Identities
1sin2 2 1coscos2 2
Homework
READ section 1.6 – IT WILL HELP!!
Pg. 57 # 1 - 75 odd