Ch 7 – Trigonometric Identities and Equations
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Transcript Ch 7 – Trigonometric Identities and Equations
Basic Trigonometric
Identities and
Equations
Trigonometric Identities
Quotient Identities
sin
tan
cos
cos
cot
sin
Reciprocal Identities
1
sin
csc
1
cos
sec
1
tan
cot
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin2 = 1 - cos2
tan2 = sec2 - 1
cot2 = csc2 - 1
cos2 = 1 - sin2
Where did our pythagorean identities come from??
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!
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
another
Pythagorean
Identity
Reciprocal
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
a third
Pythagorean
Identity
Reciprocal
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.
sin 2 cos2 1
1
1
4
sec
cos 3
3
4
2
3
sin 2 1
5
25 9
sin 2
25 25
16
2
sin
25
4
sin
5
csc
1
1
5
sin 4
4
5
3.) sinθ = -1/3, find tanθ
tan2 1 sec2
tan2 1 (3) 2
tan 2 2
tan2 8
tan2 8
4.) secθ = -7/5, find sinθ
Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
a)
cos sin tan
sin
cos sin
cos
sin 2
cos
cos
cos sin
cos
1
cos
2
sec
2
cot 2
1 sin 2
b)
cos 2
2
s
in
2
cos
1
cos 2
1
2
sin cos 2
1
sin2
csc 2
Simplifing Trigonometric Expressions
c)
(1 + tan
x)2
- 2 sin x sec x
1
(1 tanx) 2 sinx
cosx
sinx
2
1 2 tanx tan x 2
cosx
1 tan 2 x 2tan x 2 tan x
2
sec 2 x
d)
cscx
tanx cotx
1
s inx
s inx cosx
cos x s inx
1
s inx
2
2
s in x cos x
s inxcos x
1
s inx
1
s inx cosx
1
sinx cosx
sinx
1
cosx
Simplify each expression.
1
cosx
cosx
sin x
sin x
sin
cos
sin
1 sin x
cosx
sin x cosx
1
sin
sin cos
1
1
sec
cos
cos2 x sin 2 x
sin x
sin x
cos2 x sin 2 x
sin x
1
csc x
sin x
Simplifying trig Identity
Example1: simplify
tanxcosx
sin
x
tanx cosx
cos x
tanxcosx = sin x
Simplifying trig Identity
Example2: simplify
sec x
csc x
1
cos
sec x
csc
1x
sin x
=
1
sinx
x
cos x
1
=
sin x
cos x
= tan x
Simplifying trig Identity
Example2: simplify
cos2x - sin2x
cos x
cos2x - sin
1 2x
cos x
= sec x
Example
Simplify:
= cot x (csc2 x - 1)
Factor out cot x
= cot x (cot2 x)
Use pythagorean identi
= cot3 x
Simplify
Example
Simplify:
= sin x (sin x) + cos x
cos x
cos
x
2
= sin x + (cos x)cos x
cos x
= sin2 x + cos2x
cos x
=
1
cos x
= sec x
Use quotient identity
Simplify fraction with
LCD
Simplify numerator
Use pythagorean iden
Use reciprocal identity
Your Turn!
Combine
fraction
Simplify the
numerator
Use
pythagorean
identity
Use Reciprocal
Identity
Practice
1
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:
substitute using each
identity
sin x
tan x
cos x
tan x csc x
Simplify:
sec x
simplify
sin x 1
cos x sin x
1
cos x
1
cos x
1
cos x
1
1
csc x
sin x
1
sec x
cos x
Another way to use identities is to write one function in terms of another
function. Let’s see an example of this:
Write the following expression
in terms of only one trig function:
cos x sin x 1
2
= 1 sin 2 x sin x 1
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.
= sin 2 x sin x 2
sin 2 x cos2 x 1
cos2 x 1 sin 2 x
(E) Examples
• Prove tan(x) cos(x) = sin(x)
LS t an x cos x
sin x
LS
cos x
cos x
LS sin x
LS RS
20
(E) Examples
• Prove tan2(x) = sin2(x) cos-2(x)
RS sin 2 x co s2 x
1
2
RS sin x
2
co
s
x
1
2
RS sin x
co s x 2
RS
sin x 2
co s x 2
sin x
RS
co s x
RS t an2 x
2
RS LS
21
(E) Examples
• Prove
tan x
1
1
tan x sin x cos x
1
tan x
sin x
1
sin x
cos x
cos x
sin x
cos x
cos x
sin x
sin x sin x cos x cos x
cos x sin x
2
sin x cos2 x
cos x sin x
1
cos x sin x
RS
LS tan x
LS
LS
LS
LS
LS
LS
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(E) Examples
• Prove
sin 2 x
1 cos x
1 cos x
sin 2 x
LS
1 cos x
1 cos2 x
LS
1 cos x
(1 cos x )(1 cos x )
LS
(1 cos x )
LS 1 cos x
LS RS
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