12 x1 t05 02 inverse trig functions (2012)
TRANSCRIPT
Inverse Trigonometric Functions
Inverse Trigonometric Functionsxy 1sin
Inverse Trigonometric Functionsxy 1sin
xy sinxy sin y
1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range: 11 y
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range: 11 y
NO INVERSE
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range: 11 y
NO INVERSERestricted Domain:
22
x
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range: 11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range:
xyf 11 sin:
11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range:
xyf 11 sin:
Domain:
Range:
11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
11 x
22
y
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range:
xyf 11 sin:
Domain:
Range:
11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
11 x
22
y
xy sin y1
-1
22 x
Inverse Trigonometric Functionsxy 1sin
xy sinDomain: all real x
Range:
xyf 11 sin:
Domain:
Range:
11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
11 x
22
y
xy sin y1
-1
22 xy
x-1 1
2
2
Inverse Trigonometric Functionsxy 1sin
xy sin
xy 1sin
Domain: all real x
Range:
xyf 11 sin:
Domain:
Range:
11 y
NO INVERSERestricted Domain:
Range: 11 y22
x
11 x
22
y
xy sin y1
-1
22 xy
x-1 1
2
2
xy 1cos
xy 1cos xy cos
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range: 11 y
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range: 11 y
NO INVERSE
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range: 11 y
NO INVERSE
Restricted Domain: x0
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range: 11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range:
xyf 11 cos:
11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range:
xyf 11 cos:
Domain:
Range:
11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
11 x y0
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range:
xyf 11 cos:
Domain:
Range:
11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
11 x y0
xy cos
y1
-1
22 x
xy 1cos xy cos
Domain: all real x
Range:
xyf 11 cos:
Domain:
Range:
11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
11 x y0
xy cos
y1
-1
22 x
y
x-1 1
2
xy 1cos xy cos
xy 1cos
Domain: all real x
Range:
xyf 11 cos:
Domain:
Range:
11 y
NO INVERSE
Restricted Domain:
Range: 11 y
x0
11 x y0
xy cos
y1
-1
22 x
y
x-1 1
2
xy 1tan
xy 1tan xy tan
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real yintegeran is
where2
except real all
k
kx
x
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real yintegeran is
where2
except real all
k
kx
x
NO INVERSE
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real yintegeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
22
x
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real yintegeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
Range: all real y22
x
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real y
xyf 11 tan:
integeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
Range: all real y22
x
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real y
xyf 11 tan:
Domain: all real x
Range:
integeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
Range: all real y22
x
22
y
xy tan
y
x
xy 1tan xy tan
Domain:
Range: all real y
xyf 11 tan:
Domain: all real x
Range:
integeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
Range: all real y22
x
22
y
xy tan
y
x
y
x
2
2
xy 1tan xy tan
xy 1tan
Domain:
Range: all real y
xyf 11 tan:
Domain: all real x
Range:
integeran is
where2
except real all
k
kx
x
NO INVERSERestricted Domain:
Range: all real y22
x
22
y
xy tan
y
x
y
x
2
2
xx 11 sinsin (odd function)
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
43
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
43
12
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
43
12
21sin
21sin 11ii
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
43
12
21sin
21sin 11ii
64
xx 11 sinsin (odd function)
xx 11 coscos
2 shiftedfunction odd
xx 11 tantan (odd function)
2cossin 11
xx sum
e.g. Find the exact value of;
1tan3tan 11 i
43
12
21sin
21sin 11ii
64
125
4140sincos 1iii
4140sincos 1iii
4140sin 1
4041
4140sincos 1iii
4140sin 1
4041
9
4140sincos 1iii
4140sin 1
4041
9
419
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
53coslet 1
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
3
553coslet 1
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
3
545
3coslet 1
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
53
542
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
53
542
2524
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
53
542
3cos2cos 1 vi
2524
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
53
542
3cos2cos 1 vi
212cos 1
2524
4140sincos 1iii
4140sin 1
4041
9
419
6
5sinsin 1 iv6
53cos2sin 1v
cossin2 3
545
3coslet 1
53
542
3cos2cos 1 vi
212cos 1
01cos 1
25
24
41cos
32sintan 11vii
41cos
32sintan 11vii
32sinlet 1
41cos
32sintan 11vii
32sinlet 1
2 3
5
41cos
32sintan 11vii
32sinlet 1
2 3
5
1
415
41coslet 1
41cos
32sintan 11vii
32sinlet 1
2 3
5
1
415
41coslet 1
tantan1tantan
41cos
32sintan 11vii
32sinlet 1
2 3
5
1
415
41coslet 1
tantan1tantan
155
21
155
2
41cos
32sintan 11vii
32sinlet 1
2 3
5
1
415
41coslet 1
tantan1tantan
155
21
155
2
1525352
51525
5352
41cos
32sintan 11vii
32sinlet 1
2 3
5
1
415
41coslet 1
tantan1tantan
155
21
155
2
1525352
51525
5352
Exercise 1B; 1 to 5 ace etc, 6, 7ac, 8ac,9, 10ac, 11, 12ac, 14 to 20 evens