12X1 T05 02 inverse trig functions (2010)

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