Earlier we defined: [br][math]cosec\left(x\right)=\frac{1}{sin\left(x\right)},[/math][math]sec\left(x\right)=\frac{1}{cos\left(x\right)},[/math][math]cot\left(x\right)=\frac{1}{tan\left(x\right)}[/math][br]To sketch these graphs we need to find the values of the functions when x = 0, x=[math]\frac{\pi}{2}[/math], x= [math]\pi[/math], x= [br][math]\frac{3\pi}{2}[/math], x=[math]2\pi[/math][br]Then we need to find any asymptotes.[br]The functions have been sketched using GeoGebra below for you.
For the following questions, think about the following properties: domain, range, period, symmetry, intervals of increase and decrease, vertical asymptotes, y-intercepts and x-intercepts
What properties do you know about the function cosec (x)?
By looking at the graph of the function f(x) = cosec(x) we can find the following properties.[br][br][b]Domain:[/b] all real numbers except [math]k\cdot\pi[/math], as k is an integer[br][b]Range:[/b] ([math]-\infty[/math], -1] U [1, +[math]\infty[/math])[br][b]Period:[/b] [math]2\pi[/math][br][b]Vertical asymptotes:[/b] [math]x=k\cdot\pi[/math], where k is an integer
What properties do you know about the function sec (x)?
By looking at the graph of the function g(x) = sec(x) we can find the following properties.[br][br][b]Domain:[/b] all real numbers except [math]\frac{\pi}{2}+k\cdot\pi[/math], k is an integer[br][b]Range:[/b] (-[math]\infty[/math], -1] U [1, +[math]\infty[/math])[br][b]Period:[/b] 2[math]\pi[/math][br][b]Vertical asymptotes:[/b] x =[math]\frac{\pi}{2}+k\cdot p[/math], where k is an integer
What properties do you know about the function cot (x)?
By looking at the graph of the function h(x) = cot(x) we can find the following properties.[br][b]Domain:[/b] all real numbers except[math]k\cdot\pi[/math], k is an integer[br][b]Range:[/b] all real numbers[br][b]Period:[/b] [math]\pi[/math][br][b]Vertical asymptotes:[/b] [math]x=k\cdot,\pi[/math] where k is an integer