The Trigonometric Function
This is an alternative assessment task for the HSC topic the trigonometric function. This book allows students to understand and review the concepts of * (13.2) The functions sin x, cos x, tan x, cosec x, sec x, cot x and their graphs. * (13.3) Periodicity and other simple properties of the functions sin x, cos x and tan x Students are to complete each aspect in the 5 following exercises by answering all the questions.
Table of Contents
- The functions sin x, cos x, tan x and their graphs
- The functions sin x, cos x and tan x
- The functions cosec x, sec x, cot x and their graphs
- The functions cosec x, sec x and cot x
- Periodicity and other simple properties of the functions sin x, cos x and tan x
- (sin x) - Periodicity and other simple properties of functions
- (cos x) - Periodicity and other simple properties of functions
- (tan x) - Periodicity and other simple properties of functions
The functions sin x, cos x and tan x
The functions sin x, cos x, tan x
In exercise 5.7 we looked at the functions sin x, cos x, tan x and their graphs. In this GeoGebra you are going to look at how the functions can alter and what properties such as amplitude and periodicity they have.
For the following questions think about the following properties: domain, range, period, x-intercepts, y-intercepts, maximum points, minimum points, symmetry, intervals of increase and decrease and asymptotes
The function sin x
What properties do you know about the function f(x) = sin(x)?
The function cos x
What properties do you know about the function g(x)= cos (x)?
The function tan x
What properties do you know about the function h(x)= tan (x)?
The functions cosec x, sec x and cot x
The functions of cosec x, sec x, and cot x
In exercise 5.7 we briefly looked at how we can sketch the graphs of the reciprocal trigonometric functions. We can do this by finding the reciprocals of sin x, cos x and tan x at important points on the graphs. Where [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.
The graphs of the functions cosec (x), sec (x) and cot (x)
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
The function cosec x
What properties do you know about the function cosec (x)?
The function sec x
What properties do you know about the function sec (x)?
The function cot x
What properties do you know about the function cot (x)?
(sin x) - Periodicity and other simple properties of functions
In exercise 5.7 and 5.8 you looked and the periodicity and other simple properties of functions. [br]Where,[br][math]y=sin\left(x\right)[/math] has amplitude 1 and period [math]2\pi[/math][br][math]y=cos\left(x\right)[/math] has amplitude 1 and period [math]2\pi[/math][br][math]y=tan\left(x\right)[/math] has no amplitude and period [math]\pi[/math][br][br]In this GeoGebra you are going to look at the different properties and periodicity of each function and what happens when you alter the functions with different functions
Periodicity and other simple properties of f(x) = sin (x)
[math]y=a\cdot sin\left(bx\right)[/math] has amplitude [math]a[/math] and period [math]\frac{2\pi}{b}[/math][br]Have a play with the GeoGebra below. Start to notice what happens when you move the sliders
Amplitude of the function f(x)= sin(x).
Which slider in the GeoGebra changes the amplitude of the function?
Period of the function f(x) = sin(x)
Which slider in the GeoGebra changes the period of the function?
Amplitude of the function f(x) = sin(x)
Explain what happens to the amplitude of the function f(x) = sin(x) when the amplitude is 5
Amplitude of the function f(x) = sin(x)
Explain what happens to the amplitude of the function f(x) = sin(x) when the amplitude is -2.
Period of the function f(x) = sin(x)
Explain what happens to the function f(x) = sin(x) when the period (b) is [math]\frac{2\pi}{3}[/math]. (Hence the slider =3)
Period of the function f(x) = sin(x)
Explain what happens to the function f(x) when the period(b) is [math]\frac{2\pi}{-1}[/math]. (Hence the slider is -1)