6. Trigonometric Functions

Question 1

sec (x)

Answer 1

1/cos(x)

hyp/adj

Question 2

cosec(x)

Answer 2

1/sin(x)

hyp/opp

Question 3

cot(x)

Answer 3

1/tan(x)
cos(x)/sin(x)
adj/opp

Question 4

Graph of y = cosec(x)

Answer 4

Vertical asymptotes at integer coefficients of π
Passes through (π/2, 1), (π/6,2) and (5π/6,2)
Sign alternates at each asymptote
Quadratic like curves with bases at 1 and -1

Question 5

Graph of y = sec(x)

Answer 5

Vertical asymptotes at integer.5 coefficients of π
Passes through (0, 1), (π/3,2) and (-π/3,2)
Sign alternates at each asymptote
Quadratic like curves with bases at 1 and -1

Question 6

Graph of y = cot(x)

Answer 6

Vertical asymptotes at integer coefficients of π
Starts high to the right of the asymptotes, flattens and passes through 0 at integer.5 coefficients of π and then falls at a steeper gradient at the bottom

Question 7

Reciprocal trig identities

Answer 7

sec^2 x = 1 + tan^2 x
cosec^2 x = 1 + cot^2 x
Derive by dividing sin^2 x + cos^2 x = 1 by cos^2 x and sin^2 x respective locations

Question 8

arcsin, arccos, arctan

Answer 8

The inverse trig functions

Question 9

Graph of y = arcsin(x)

Answer 9

x values from -1 to 1
y values from -90 to 90
(-1,-90), rises with a steep gradient, diagonally through the origin and steeper again to (1,90)
Alternate y and x values from sin graph

Question 10

Graph of y = arccos(x)

Answer 10

x values from -1 to 1
y values from 180 to 0
(-1,180), falls with a steep gradient, diagonally through (0,90) and steeper fall again to (1,0)
Alternate y and x values from cos graph

Question 11

Graph of y = arctan(x)

Answer 11

x values all real values
y values from -90 to 90 with asymptotes
Almost horizontal at -90 and 90 with a steep rise between through the origin