7.1-7.2 basic trigonometric identities
DESCRIPTION
7.1-7.2 Basic Trigonometric Identities In this powerpoint, we will use trig identities to verify and prove equations. See what you get. Etc. Proving an Identity. Prove the following:. a) sec x (1 + cos x ) = 1 + sec x. = sec x + sec x cos x = sec x + 1. 1 + sec x. - PowerPoint PPT PresentationTRANSCRIPT
7.1-7.2 Basic Trigonometric Identities
In this powerpoint, we will use trig identities to verify and prove equations
cos
sin1
2 2sin cos 1
1 1 sin cossec csc tan cot
cos sin cos sin
1cot
tan
2 2sin cos 1
2 2
2 2 2
sin cos 1
cos cos cos
2 2
2 2 2
sin cos 1
sin sin sin
2 2tan 1 sec
2 21 cot csc
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
sin cot tan sec
cos sinsin sin sec
sin cos
2sin
cos seccos
2 2cos sin
seccos cos
1
seccos
sec sec
€
cosθ
cosθ
⎛
⎝ ⎜
⎞
⎠ ⎟•
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
2 21 cos 1 cot 1 2 2sin csc 1
22
1sin 1
sin
1 1
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
sincsc cot
1 cos
1 cos sin
sin sin 1 cos
1 cos
1 co
1 cos sin
sin 1 coss
21 cos sin
sin 1 cos 1 cos
2sin sin
sin 1 cos 1 cos
sin sin
1 cos 1 cos
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
csc 1 1 sin
csc 1 1 sin
sin
sin
csc 1 1 sin
csc 1 1 sin
1 sin 1 sin
1 sin 1 sin
See what you get
1 1 sin cos 1sec csc tan cot cot
cos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
cos 1 sin2sec
1 sin cos
22cos 1 sin2sec
cos 1 sin
2 2cos 1 2sin sin
2seccos 1 sin
2 1 sin2sec
cos 1 sin
2
2seccos
2sec 2sec
21 coscsc cot
1 cos
21 cos 1 coscsc cot
1 cos 1 cos
2
2
2
1 coscsc cot
sin
2
21 coscsc cot
sin
2 2csc cot csc cot
€
(1
sin x−
cos x
sin x)2
2 2 2
sin cos tan
cos sin 1 tan
2
2
2 2
2 sin cos ta1
cos1
c
n
cos sin 1 tanos
2 2
2
sintancos
sin 1 tan1
cos
2 2
tan tan
1 tan 1 tan
22
2
1 cot2cos 1
1 cot
2
22
1 cot2cos 1
csc
2
22 2
1 cot2cos 1
csc csc
2 2 2sin cos 2cos 1
2 2sin cos 1 1 1
1 sin 1 sin4 tan sec
1 sin 1 sin
2 2
2
1 sin 1 sin4 tan sec
1 sin
2 2
2
1 2sin sin 1 2sin sin4 tan sec
cos
4sin
4 tan seccos cos
4 tan sec 4 tan sec
3 3
2
sin cos sec sin
tan 11 2cos
2 2
2
sin cos sin sin cos cos sec sin
tan 11 2cos
2
sin cos 1 sin cos sec sin
tan 11 2cos
2 2 2
sin cos 1 sin cos sec sin
tan 1sin cos 2cos
2 2
sin cos 1 sin cos sec sin
tan 1sin cos
sin cos 1 sin cos sec sin
sin cos sin cos tan 1
11 sin cos sec sincos
1 tan 1sin coscos
Etc.
5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
=sinxcosx
×1
sinx
=1
cosx
=secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x
=sinxcosx
×sinx
1+cosx
=sin2x+cos2x
cosx
=1
cosx=secx
secx
L.S. = R.S.
d) sin4x - cos4x = 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x)= (1 - cos2x - cos2x)= 1 - 2cos2x
L.S. = R.S.
1 - 2cos2x
e)
11+cosx
+1
1−cosx = 2csc2x
=(1−cosx)+(1+cosx)(1+cosx)(1−cosx)
=2
(1−cos2x)
=2
sin2x
=2csc2x
2csc2x
L.S. = R.S.
Proving an Identity
5.4.9
Proving an Identity
5.4.10
f)
cosA1+sinA
+1+sinAcosA
= 2secA
=cos2A+(1+sinA)(1+sinA)
(1+sinA)(cosA)
=cos2A+(1+2sinA+sin2A)
(1+sinA)(cosA)
=cos2A+sin2A+1+2sinA
(1+sinA)(cosA)
=2+2sinA
(1+sinA)(cosA)
=2(1+sinA)
(1+sinA)(cosA)
=2
(cosA)=2secA
2secA
L.S. = R.S.