Trigonometric Review

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