Basic Trigonometric Identities
cosx/sinx
1+sec^2x sin^2x = sec^2x
1+sec^2x sin^2x = sec^2x
= 1+(1/cos^2x) sin^2x = sec^2x
= 1+ (sin^2x/cos^2x) = sec^2x
= 1 + tan^2x = sec^2x
Verifies because it is a Pythagorean Identity
sin135
sin (180-45)=
sin (180) cos (45) - cos (180) sin (45)
0 * √ 2/2 - (-1) * √ 2/2
= √ 2/2
cscxsinx
cscxsinx
=(1/sinx)(sinx)
=sinx/sinx
=1
1-cos^2x=?
sin^2x
sinx secx cotx = 1
sin * (1/cosx) * (cosx / sinx )
sinx/ cosx * cosx/ sinx
=1
cos 7pi/12
cos 105
cos(45 +60)
cos (45) * cos (60) - sin (45) * sin (60)
√2/2 * 1/2 - √2/2 * √3/2
V2/4 - √ 6/4
( √2 - √6 ) / 4
cosx/secx
cosx/secx
=cosx/(1/cosx)
KCF = cosx * cosx
=cos^2x
sec^2x - 1
tan^2x
cot x ( cotx + tanx ) = csc^2x
cot^2x + cotx * tanx
cot^2x + (cosx/sinx) * (sinx/cosx)
cot^2x + 1 (Pythagorean Identity)
= csc^2x
sin pi/12
sin 15
sin (45-30)
sin (45) * cos (30) - cos (45) * sin (30)
√2/2 * √3/2 - √2/2 * 1/2
( √6 -√2 ) / 4
( 1 + tan^2x ) / csc^2x
1 + tan^2x = sec^2x
sec^2x/ csc^2x
(1/cos^2x) / (1/sin^2x)
KCF : (1/cos^2x) * (sin^2x)
sin^2x / cos^2x = tan^2x
sinx / cosx
tanx
( 1- 2cos^2x ) / ( sinx cosx ) = tanx - cotx
Pythagorean identity: sin^2x +cos^2x +1
replace 1 with
( sin^2x +cos^2x ) - 2cos^2x
( sin^2x - cos^2x ) / ( sinxcosx)
( sin^2x / sinx cosx ) - ( cos^2x / sinx cosx )
( sinx/ cosx ) - ( cosx /sinx )
tanx - cotx
Verify : cos ( 90 - x) = sinx
cos 90 * cos x + sin 90 * sin x
0 * cos x + 1 * sin x
= sinx
sinx (cotx - cscx)
sinx cotx - sinx cscx (distribute)
sinx (cosx/sinx) - sinx (1/sinx)
cosx - 1
csc^2x - cot^2x
1
cot^2x / (1 + cscx ) = ( 1- sinx) / sinx
cot^2x = csc^2x -1
( csc^2x -1 ) / ( 1 +cscx )
We can factor the numerator into :
( cscx +1 ) (cscx -1 ) / ( 1+cscx)
cscx -1 = ( 1 -sinx) / sinx
cscx -1 = ( 1/ sinx ) - ( sinx/ sinx )
1/sinx = csc x -1
tan 195
tan (135 +60)
( tan 135 + tan 60 ) / (1-tan135tan60)
( -1 + √ 3 ) / (1 - (-1)* √ 3)
( √ 3 -1 ) / ( √ 3 +1 ) * ( √ 3 -1 ) / ( √ 3 -1 )
= 2 - √ 3
( sinx ) ( 1 + cot^2x )
( sinx ) (csc^2x )
( sinx ) ( 1/sin^2x )
sin x / sin^2x
1/ sinx = cscx