7.1 – Basic Trigonometric Identities and
Equations
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
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??
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
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
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
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
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
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
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
Simplifying trig Identity
Example1: simplify tanxcosx
tanx cosxsin xcos x
tanxcosx = sin x
Example2: simplifysec xcsc x
sec xcsc x1sin x
1cos x 1
cos xsinx
1= x
=sin xcos x
= tan x
Simplifying trig Identity
Simplifying trig Identity
Example2: simplify cos2x - sin2x
cos x
cos2x - sin2x
cos xcos2x - sin2x 1 = sec x
ExampleSimplify:
= cot x (csc2 x - 1)
= cot x (cot2 x)
= cot3 x
Factor out cot x
Use pythagorean identity
Simplify
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
Your Turn!Combine fraction
Simplify the numeratorUse pythagorean identity
Use Reciprocal Identity
Practice
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
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
20
(E) Examples
• Prove tan(x) cos(x) = sin(x)
RSLS
xLS
xx
xLS
xxLS
sin
coscos
sin
costan
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
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
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