New Zealand
Level 8 - NCEA Level 3

# Domain and range of cot, sec and cosec curves

## Interactive practice questions

Consider the function $f\left(x\right)=\csc x$f(x)=cscx, which is defined as the reciprocal function of $\sin x$sinx. That is, $\csc x=\frac{1}{\sin x}$cscx=1sinx.

a

Which of the following describes the values of $x$x where $\sin x=0$sinx=0?

$x=\pi n+\frac{\pi}{2}$x=πn+π2 where $n$n is an integer.

A

$x=2\pi n$x=2πn where $n$n is an integer.

B

$x=\pi n$x=πn where $n$n is an integer.

C

$x=\frac{\pi}{2}n$x=π2n where $n$n is an integer.

D

$x=\pi n+\frac{\pi}{2}$x=πn+π2 where $n$n is an integer.

A

$x=2\pi n$x=2πn where $n$n is an integer.

B

$x=\pi n$x=πn where $n$n is an integer.

C

$x=\frac{\pi}{2}n$x=π2n where $n$n is an integer.

D
b

What does your answer to part (a) imply about the function $f\left(x\right)=\csc x$f(x)=cscx?

$f\left(x\right)$f(x) is undefined when $x=\pi n$x=πn.

A

$f\left(x\right)$f(x) will reach its maximum value when $x=\pi n$x=πn.

B

$f\left(x\right)=0$f(x)=0 when $x=\pi n$x=πn.

C

$f\left(x\right)$f(x) will reach its minimum value when $x=\pi n$x=πn.

D

$f\left(x\right)$f(x) is undefined when $x=\pi n$x=πn.

A

$f\left(x\right)$f(x) will reach its maximum value when $x=\pi n$x=πn.

B

$f\left(x\right)=0$f(x)=0 when $x=\pi n$x=πn.

C

$f\left(x\right)$f(x) will reach its minimum value when $x=\pi n$x=πn.

D
Easy
Approx 2 minutes

Consider the function $f\left(x\right)=\sec x$f(x)=secx, which is defined as the reciprocal function of $\cos x$cosx. That is, $\sec x=\frac{1}{\cos x}$secx=1cosx.

Consider the function $f\left(x\right)=\cot x$f(x)=cotx, which is defined as the reciprocal function of $\tan x$tanx. That is, $\cot x=\frac{1}{\tan x}$cotx=1tanx.

Consider the function $f\left(x\right)=\sec x$f(x)=secx, which is defined as the reciprocal function of $\cos x$cosx. That is, $\sec x=\frac{1}{\cos x}$secx=1cosx.

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions