# ACT.trig.01 What is the period of the function csc 4x

• MHB
• karush
In summary, the function csc 4x represents the cosecant of 4 times x, and its period is equal to π/2. This is different from other trigonometric functions because it is the reciprocal of the period of sin 4x. The domain of csc 4x is all real numbers except for values that make the function undefined, and the range is all real numbers except for 0. Csc 4x is related to other trigonometric functions through identities and can be expressed in terms of other functions. Its graph is also a reflection of the graph of sec 4x across the y-axis.

#### karush

Gold Member
MHB
$\tiny{ACT.trig.01}$
What is the period of the function $f(x)=\csc{4x}$
$a. \pi \quad b, 2\pi \quad c. 4\pi \quad d. \dfrac{\pi}{4} \quad e. \dfrac{\pi}{2}$

well we should know the answer by observation
but I had to graph it
looks like $\dfrac{\pi}{2}$

$y = \csc(Bx)$$T = \dfrac{2\pi}{B} = \dfrac{2\pi}{4} = \dfrac{\pi}{2}$

Hard to remember stuff like that
my mind freezes at tests alto I got some A's occasionally:unsure:

## 1. What does the function csc 4x represent?

The function csc 4x represents the cosecant of 4 times x, where x is the input value.

## 2. What is the period of the function csc 4x?

The period of the function csc 4x is the length of one full cycle of the graph. In this case, the period is equal to 2π/4, or π/2. This means that the graph of csc 4x repeats every π/2 units.

## 3. How is the period of csc 4x different from other trigonometric functions?

The period of csc 4x is different from other trigonometric functions because it is the reciprocal of the period of sin 4x. This means that the graph of csc 4x will have more peaks and valleys than the graph of sin 4x within the same interval.

## 4. How do you find the domain and range of csc 4x?

The domain of csc 4x is all real numbers except for values of x that make the function undefined, which occur when the input value is equal to π/2, 3π/2, 5π/2, etc. The range of csc 4x is all real numbers except for 0, since the function is undefined at those points.

## 5. What is the relationship between csc 4x and other trigonometric functions?

Csc 4x is related to other trigonometric functions through various identities, such as the Pythagorean identity and the reciprocal identities. It can also be expressed in terms of other trigonometric functions, such as csc 4x = 1/sin 4x. Additionally, the graphs of csc 4x and sec 4x are reflections of each other across the y-axis.