7.1 – Basic Trigonometric Identities and Equations
description
Transcript of 7.1 – Basic Trigonometric Identities and Equations
![Page 1: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/1.jpg)
7.1 – Basic Trigonometric Identities and
Equations
![Page 2: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/2.jpg)
5.4.3
Trigonometric Identities
Quotient Identities
tanθ=sinθcosθ
cotθ=cosθsinθ
Reciprocal Identities
sinθ=1
cscθcosθ=
1secθ
tanθ=1
cotθ
Pythagorean Identities
sin2+ cos2 = 1 tan2+ 1 = sec2 cot2+ 1 = csc2
sin2= 1 - cos2
cos2 = 1 - sin2
tan2= sec2- 1 cot2= csc2- 1
![Page 3: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/3.jpg)
Do you remember the Unit Circle?
• What is the equation for the unit circle?x2 + y2 = 1
• What does x = ? What does y = ? (in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean Identity!
Where did our pythagorean identities come from??
![Page 4: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/4.jpg)
Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ
Quotient Identity
ReciprocalIdentityanother
Pythagorean Identity
![Page 5: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/5.jpg)
Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ
Quotient Identity
ReciprocalIdentitya third
Pythagorean Identity
![Page 6: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/6.jpg)
Using the identities you now know, find the trig value.
1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.
€
secθ =1
cosθ=
13
4=
4
3
€
sin2θ + cos2θ =1
sin2θ +3
5
⎛
⎝ ⎜
⎞
⎠ ⎟2
=1
sin2θ =25
25−
9
25
sin2θ =16
25
sinθ = ±4
5
cscθ =1
sinθ=
1
± 45
= ±5
4
![Page 7: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/7.jpg)
3.) sinθ = -1/3, find tanθ
4.) secθ = -7/5, find sinθ€
tan2θ +1 = sec2θ
tan2θ +1 = (−3)2
tan2θ = 8
tan2θ = 8
€
tanθ = 2 2
![Page 8: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/8.jpg)
Identities can be used to simplify trigonometric expressions.
Simplifying Trigonometric Expressions
cosθ+sinθ tanθ
=cosθ +sinθ
sinθcosθ
=cosθ +
sin2θcosθ
=
cos2θ + sin2θcosθ
=1
cosθ
=secθ
a)
Simplify.
b)cot2θ
1−sin2θ
=
cos2θsin2θcos2θ
1
=1
sin2θ
=csc2θ
5.4.5
=cos2θsin2θ
×1
cos2θ
![Page 9: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/9.jpg)
Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x
=1+2tanx+tan2x−2sinxcosx
=1+tan2x+2tanx−2tanx
=sec2x
d)cscx
tanx+cotx
=1
sinxsinxcosx
+cosxsinx
=1
sinxsin2x+cos2x
sinxcosx
=1
sinx×
sinxcosx1
=cosx
=1
sinx1
sinxcosx
=(1+tanx)2 −2sinx1
cosx
![Page 10: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/10.jpg)
Simplify each expression.
€
1sinθ
cossinθ
1
sinθ•
sinθ
cosθ
1
cosθ= secθ
€
=cos x1
sin x
⎛
⎝ ⎜
⎞
⎠ ⎟sin x
cos x
⎛
⎝ ⎜
⎞
⎠ ⎟
=1
€
cos xcos x
sin x
⎛
⎝ ⎜
⎞
⎠ ⎟+ sin x
cos2 x
sin x+
sin2 x
sin x
cos2 x + sin2 x
sin x
1
sin x= csc x
![Page 11: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/11.jpg)
Simplifying trig Identity
Example1: simplify tanxcosx
tanx cosxsin xcos x
tanxcosx = sin x
![Page 12: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/12.jpg)
Example2: simplifysec xcsc x
sec xcsc x1sin x
1cos x 1
cos xsinx
1= x
=sin xcos x
= tan x
Simplifying trig Identity
![Page 13: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/13.jpg)
Simplifying trig Identity
Example2: simplify cos2x - sin2x
cos x
cos2x - sin2x
cos xcos2x - sin2x 1 = sec x
![Page 14: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/14.jpg)
ExampleSimplify:
= cot x (csc2 x - 1)
= cot x (cot2 x)
= cot3 x
Factor out cot x
Use pythagorean identity
Simplify
![Page 15: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/15.jpg)
ExampleSimplify:
Use quotient identity
Simplify fraction with LCD
Simplify numerator
= sin x (sin x) + cos xcos x
= sin2 x + (cos x)cos x
cos xcos x
= sin2 x + cos2x
cos x = 1
cos x
= sec x
Use pythagorean identity
Use reciprocal identity
![Page 16: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/16.jpg)
Your Turn!Combine fraction
Simplify the numeratorUse pythagorean identity
Use Reciprocal Identity
![Page 17: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/17.jpg)
Practice
![Page 18: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/18.jpg)
One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:
sintan
cos
xx
x
1sec
cosx
x
1csc
sinx
x
tan cscSimplify:
sec
x x
x
sin 1cos sin
1cos
xx x
x
substitute using each identity
simplify
1cos
1cos
x
x
1
![Page 19: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/19.jpg)
Another way to use identities is to write one function in terms of another function. Let’s see an example of this:
2
Write the following expression
in terms of only one trig function:
cos sin 1x x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.
2 2sin cos 1x x 2 2cos 1 sinx x
2= 1 sin sin 1x x
2= sin sin 2x x
![Page 20: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/20.jpg)
20
(E) Examples
• Prove tan(x) cos(x) = sin(x)
RSLS
xLS
xx
xLS
xxLS
sin
coscos
sin
costan
![Page 21: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/21.jpg)
21
(E) Examples
• Prove tan2(x) = sin2(x) cos-2(x)
LSRS
xRS
x
xRS
x
xRS
xxRS
xxRS
xxRS
2
2
2
2
2
2
2
2
22
tan
cos
sin
cos
sin
cos
1sin
cos
1sin
cossin
![Page 22: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/22.jpg)
22
(E) Examples
• Prove tan
tan sin cosx
x x x
1 1
LS xx
LSx
x xx
LSx
x
x
x
LSx x x x
x x
LSx x
x x
LSx x
LS RS
tantan
sin
cos sincos
sin
cos
cos
sinsin sin cos cos
cos sin
sin cos
cos sin
cos sin
1
1
1
2 2
![Page 23: 7.1 – Basic Trigonometric Identities and Equations](https://reader035.fdocuments.in/reader035/viewer/2022081516/56813af5550346895da37979/html5/thumbnails/23.jpg)
23
(E) Examples
• Prove sin
coscos
2
11
x
xx
LSx
x
LSx
x
LSx x
x
LS x
LS RS
sin
cos
cos
cos( cos )( cos )
( cos )
cos
2
2
1
1
11 1
1
1