Trigonometric Identities
What s an identity?
An equation that is true for all values of the
variable for which both sides of the equation
are defined.
We have already established a bunch of identities from
the last chapter.
RECIPROCAL _ IDENTITIES
1
sin θ
1
sec θ =
cosθ
1
cot θ =
tan θ
csc θ =
QUOTIENT _ IDENTITIES
sin θ
tan θ =
cosθ
cosθ
cot θ =
sin θ
Based on whether a function has origin or yaxis symmetry we have EVEN-ODD identities
as well.
EVEN − ODD _ IDENTITIES
sin( −θ ) = − sin θ
Origin symmetry
csc( −θ ) = − csc θ
EVEN − ODD _ IDENTITIES
cos( −θ ) = cosθ
Y-axis symmetry
sec( −θ ) = sec θ
EVEN − ODD _ IDENTITIES
tan( −θ ) = − tan θ
cot( −θ ) = − cot θ
Origin symmetry
Pythagorean Theorem from
the unit circle yields the
first Pythagorean identity.
Pythagorean Identities
Begin with the unit circle
(cos! , sin! )
1
sin θ
cosθ
cos θ + sin θ = 1
2
2
cos θ + sin θ = 1
2
2
To get the second identity, divide each term by cosine
squared theta.
cos θ sin θ
1
+
=
2
2
2
cos θ cos θ cos θ
2
2
Identity #2
Simplify…..
1 + tan θ = sec θ
2
2
cos θ + sin θ = 1
2
2
To get the last Pythagorean Identity, divide each term
by sine squared theta.
cos2 θ sin 2 θ
1
+ 2 = 2
2
sin θ sin θ sin θ
Simplify….
cot θ + 1 = csc θ
2
2
Applying these identities to a situation.
Use identities to find the exact values of the five other
trig functions given….
8
sin θ = _ & _ cosθ < 0
17
sin θ _ is_ positive
cos ineθ _ is _ negative
So, must _ be _ 2 ND _ QUADRANT _ angle
cos2 θ + sin 2 θ = 1
cosθ = ± 1 − sin 2 θ
Applying these identities to a situation.
Use identities to find the exact values of the five other
trig functions given….
8
sin θ = _ & _ cosθ < 0
17
You are given that cosine is negative
" 8%
cos! = ! 1 ! $ '
# 17 &
2
" 8%
cos! = ! 1 ! $ '
# 17 &
2
64
cosθ = − 1 −
289
225 −15
cosθ = −
=
289 17
Find the other 3 trig functions by
using reciprocal identities.
" 8%
cos! = ! 1 ! $ '
# 17 &
2
−15
cot θ =
8
−17
sec θ =
15
17
csc θ =
8
Simplifying a trig expression using the Pythagorean
Identities
Simplify:
cot 2 t sin 2 t + sin 2 t
Note:
2
cos
t
2
cot t =
sin 2 t
Substitution :
! cos2 t $ 2
2
sin
t
+
sin
t
#" sin 2 t &%
Simplifying a trig expression using the Pythagorean
Identities
Simplify:
cot 2 t sin 2 t + sin 2 t
Simplify:
cos t + sin t = 1
2
2
Simplify the following equation:
Note:
cos( −θ ) = cos(θ )
Note:
csc 2 (θ ) − 1 = cot 2 (θ )
cot θ cos( −θ )
2
csc θ − 1
cot θ cos(θ )
2
csc θ − 1
cot θ cos(θ )
2
cot θ
Simplify the following equation:
cot θ cos( −θ )
2
csc θ − 1
Reduce
cos(θ )
cot θ
Note:
cosθ
cot θ =
sin θ
Simplify:
cos(θ )
= sin θ
cosθ
sin θ
Verify the following identity:
(cos t )(1 + tan t ) = 1
2
Note:
1 + tan 2 t = sec 2 t
Note:
1
sec t =
2
cos t
2
2
(cos t )(sec t ) = 1
2
(
2
! 1 $
cos t #
=1
2 &
" cos t %
1=1
2
)
Verify a trickier one:
Note:
cos t
1 + sin t
=
1 − sin t
cos t
1 − sin 2 t = cos2 t
(1 + sin t ) cost = 1 + sin t
(1 + sin t ) 1 ! sin t cost
(1 + sin t ) cost = 1 + sin t
1 ! sin 2 t
cost
Verify a trickier one:
cos t
1 + sin t
=
1 − sin t
cos t
(1 + sin t ) cost = 1 + sin t
2
cos t
cost
(1 + sin t ) = 1 + sin t
cost
cost