Year 2 Chapter 6: Trigonometric Functions

Question 1

What are sec x, cosec x, and cot x equivalent to (give 2 things that cot x is equivalent to)?

Answer 1

. Sec x= 1/cos(x).
. Cosec x= 1/sin(x).
. Cot x= 1/tan(x) = cos(x)/sin(x)

Question 2

Give 3 asymptotes for the graph of y=sec (x), y=cosec (x) and y=cot (x):

Answer 2

. Y=sec (x): x=-90°, x=90°, x=270°.
. Y= cosec (x): x=-180°, x=0°, x=180°.
. Y= cot (x): x=-180°, x=0°, x=180°.

Question 3

Give the y-i for the graph of y=sec (x), y=cosec (x) and y=cot (x):

Answer 3

. Y=sec (x): 1
. Y= cosec (x): ‘X=0’ is an asymptote.
. Y= cot (x): ‘X=0’ is an asymptote.

Question 4

Give the domain for the graph of y=sec (x), y=cosec (x) and y=cot (x):

Answer 4

. Y=sec (x): x E R, x≠-90°, 90°, 270°….
. Y= cosec (x): x E R, x≠-180°, 0°, 180°….
. Y= cot (x): x E R, x≠-180°, 0°, 180°….

Question 5

Give the period for the graph of y=sec (x), y=cosec (x) and y=cot (x):

Answer 5

. Y=sec (x): 360°.
. Y= cosec (x): 360°.
. Y= cot (x): 180°.

Question 6

Give the x-i for the graph of y=sec (x), y=cosec (x) and y=cot (x):

Answer 6

. Y=sec (x): no values.
. Y= cosec (x): no values
. Y= cot (x): x= 90°+180k°, where k is a whole number (k can be negative so x can be -90°).

Question 7

Briefly summarise what y=sec(x) looks like:

Answer 7

. An infinite series of U and n shapes.
. All U shapes are over x-axis and the same shape as each other.
. All n shapes are under x-axis and the same shape as each other.
. The U shape that goes through y-axis is between 2 asymptotes.
- Then there is an n-shape between the next 2 asymptotes, then a u-shape between the next 2 asymptotes, then an n-shape between the next 2, and so on….

Question 8

Briefly summarise what y=cosec(x) looks like:

Answer 8

. An infinite series of U and n shapes.
. All U shapes are over x-axis and the same shape as each other.
. All n shapes are under x-axis and the same shape as each other.
. The U shape that is closest to y-axis is between 2 asymptotes.
- Then there is an n-shape between the next 2 asymptotes, then a u-shape between the next 2 asymptotes, then an n-shape between the next 2, and so on….

Question 9

Briefly summarise what y=cot(x) looks like:

Answer 9

. An infinite series of horizontally reflected ‘y=tan x’ lines.
. All lines are infinite in vertical height and the same shape as each other.
. The ‘y=cot x’ line, that is to the left of and closest to, the y-axis is between 2 asymptotes.
- Then there is the same line between the next 2 asymptotes, then the same line between the next 2, and so on…..

Question 10

What are min and max values for y=sec(x) and y=cosec(x)?

Answer 10

. Min for both functions: 1.

. Max for both functions: -1.

Question 11

What is 1+tan²(x) equivalent to?

Answer 11

Sec²(x)

Question 12

What is 1+cot²(x) equivalent to?

Answer 12

Cosec²x

Question 13

Give the domain of y=arcsin(x), y=arccos(x) and y=arctan(x):

Answer 13

. Y=arcsin(x): x is between (but equal to) -1 and 1.
. Y=arccos(x): x is between (but equal to) -1 and 1
. Y=arctan(x): x E R.

Question 14

Give the range of y=arcsin(x), y=arccos(x) and y=arctan(x):

Answer 14

. Y=arcsin(x): y is between (but equal to) -90° and 90°
. Y=arccos(x): y is between (but equal to) 0° and 180°
. Y=arctan(x): y is between (but NOT equal to) -90° and 90°.

Question 15

Give the range of y=sec(x), y=cosec(x) and y=cot(x):

Answer 15

. Y=sec (x): y is less than (or equal to) -1 but more than (or equal to) 1.
. Y= cosec (x): y is less than (or equal to) -1 but more than (or equal to) 1.
. Y= cot (x): x E R

Question 16

Give the x-i of y=arcsin(x), y=arccos(x) and y=arctan(x):

Answer 16

. Y=arcsin(x): x=0
. Y=arccos(x): x=1
. Y=arctan(x): x=0

Question 17

Give the y-i of y=arcsin(x), y=arccos(x) and y=arctan(x):

Answer 17

. Y=arcsin(x): y=0
. Y=arccos(x): y=90°
. Y=arctan(x): y=0

Question 18

Briefly summarise what y=arcsin(x) looks like:

Answer 18

. A line that goes from vertical to straight with a medium gradient then steep again, maintaining a positive gradient throughout.
. It looks like ‘y=sin(x)’ reflected in the line y=x.

Question 19

Briefly summarise what y=arccos(x) looks like:

Answer 19

. A line that goes from vertical to straight with a medium gradient then steep again, maintaining a negative gradient and not going under the x-axis.
. It looks like ‘y=arcsin(x)’ reflected in the line y=0 then transformed by vector (0,90°).

Question 20

Briefly summarise what y=arctan(x) looks like:

Answer 20

. A line that goes from almost flat to almost vertical (around the origin) then back to almost flat.
. Looks like y=arccos(x) transformed by vector (0,-90°) then rotated 90° anti-clockwise.