In this page, we will learn about the graphs of the circular functions sinx, cosx and tanx. We will link this to work that we have done on Transforming Functions to see how translating and stretching the graphs affect the tigonometric functions. It is fairly common to get a long question on this topic in the exam and they can come up on paper 1 or paper 2, so you need to be comfortable using your calculator and working without it. You can find typical questions in the exam-style questions section.
On this page, you should learn about
- the graphs of the functions sinx, cosx and tanx
- the amplitude and period of a trigonometric graph
- Composite functions of the form \(\large f(x) = a \sin(b(x+c))+d\) and \(\large f(x) = a \cos(b(x+c))+d\)
- Transformations of trigonometric functions
- Modelling with trigonometric functions
Here is a quiz about the period of a trigonometric function
START QUIZ!
What is the period of this function?
period =
f(x) = sin 2x
What is the period of the function f ?
The function f(x) = sin(ax) has period \(\large \frac{2\pi}{a}\)
f(x) = cos 0.5x
What is the period of the function f ?
The function f(x) = cos(ax) has period \(\large \frac{2\pi}{a}\)
\(\large f(x) = \sin \pi x\)
What is the period of the function f ?
period =
The function f(x) = sin(ax) has period \(\large \frac{2\pi}{a}\)
\(\large f(x) = \cos \frac{\pi x}{2}\)
What is the period of the function f ?
period =
The function f(x) = cos(ax) has period \(\large \frac{2\pi}{a}\)
\(\large f(x) = \sin 4\pi x\)
What is the period of the function f ?
period =
The function f(x) = sin(ax) has period \(\large \frac{2\pi}{a}\)
\(\large f(x) = 2\cos \frac{\pi x}{15}\)
What is the period of the function f ?
period =
The function f(x) = cos(ax) has period \(\large \frac{2\pi}{a}\)
\(\large f(x) = \sin (\frac{2\pi }{5}(x-5))\)
What is the period of the function f ?
period =
The function \(\large f(x) = \sin (a(x-b))\) has period \(\large \frac{2\pi}{a}\)
What is the equation for this function?
The function f(x) = cos(ax) has period \(\large \frac{2\pi}{a}\)
What is the equation for this function?
The function f(x) = sin(ax) has period \(\large \frac{2\pi}{a}\)
Here is a quiz about the amplitude of a trigonometric function
START QUIZ!
What is the amplitude of this function?
amplitude =
What is the amplitude of this function?
amplitude =
What is the amplitude of this function?
amplitude =
What is the amplitude of this function?
\(\large f(x)=2\cos 3x\)
amplitude =
The function, f(x) = a cosbx has amplitude |a|
What is the amplitude of this function?
\(\large f(x)=-3\sin 0.5x\)
amplitude =
The function, f(x) = a sin bx has amplitude |a|
The diagram shows the graph of f.
What is the value of a in the following function? \(\large f(x)= a\cos(x-\frac{\pi}{3})+1\)
a =
The diagram shows the graph of f.
What is the value of a in the following function? \(\large f(x)= a\sin(5x)+4\)
a =
The amplitude = 3
However, the graph of y = 3 sin(5x) has been reflected in the x axis, so f(x) = -3 sin(5x)
Which is the correct graph of the function \(\large f(x) = -1.5\sin(x+\frac{\pi}{2})+1.5\)
| A | B | C | D |
 |  |  |  |
\(\large f(x) = -1.5\sin(x+\frac{\pi}{2})+1.5\)
Start with graph of y = sinx
- Translate left \(\large\frac{\pi}{2}\)
- Stretch by a factor 1.5 in the y direction
- Reflect in x axis
- Translate up 1.5 units
The graph of f is plotted below. Which is the correct function?
The amplitude is 3.5 and the graph of cos x has been reflected in the x axis. Hence a = -3.5
The graph of f is plotted below. Which is the correct function?
The amplitude is 2.5
a is positive
Here is a quiz about translations of a trigonometric function
START QUIZ!
The graph of f is shown below. Which is the correct function?
The graph of y = 2 sinx is translated right \(\large \frac{\pi}{2}\)
The graph of f is shown below. Which is the correct function?
The graph of y = -0.5 cosx is translated left \(\large \frac{\pi}{3}\)
The diagram below shows the graph of \(\large f(x) = 2\sin x+a\)
What is the value of a?
a =
The graph of y = 2 sinx has been translated up 1 unit
The diagram below shows the graph of \(\large f(x) = -3\cos x +a\)
What is the value of a?
a =
The graph of y = -3cosx is translated up 3 units
The diagram below shows the graph of \(\large f(x) = 0.5\sin 2x +a\)
What is the value of a?
a =
The graph of y =0.5sin2x is translated down 1.5 units
The diagram below shows the graph of \(\large f(x) =-2.5\sin(\frac{\pi}{5}x)+a\)
What is the value of a?
a =
The graph of \(\large y =-2.5\sin(\frac{\pi}{5}x)+a\) is translated up 3.5 units
The graph of f is shown below. Which is the correct function?
The graph of \(\large y = \sin\pi x\) is translated left by 1 unit
The graph of f is shown below. Which is the correct function?
The graph of \(\large y = \cos\frac{\pi}{2} x\) is translated right \(\large \frac{1}{2}\)
The graph of f is shown below. Which is the correct function?
The graph of \(\large y=-2 \cos(\frac{\pi}{5}x)\) is translated right 2
The graph of f is shown below. Which is the correct function?
The graph of \(\large y=3 \sin(\frac{\pi}{3}x)\) is translated left 1.5
The height h of water, in metres, in a habour is modelled by the function \(\large h(t)=5.5\sin(0.5(t-1.5))+12\) where t is time after midday in hours.
a) Find the initial height of the water.
b) At what time is it when the water reaches this height again?
c) Find the maximum height of the water.
d) How much time is there in between the first and second time that the water at 16 metres?
Give heights to 3 significant figures and times to the nearest minute
Hint
Full Solution
0.283x60 = 17 minutes
The following diagram shows a Ferris wheel.
The height, h metres of a seat above ground after t minutes is given by \(\large h(t)=a\ \cos(bt)+c\) , where a, b and c \(\large \in \mathbf{R}\)
The following graph shows the height of the seat.
Find the values of a, b and c
Hint
Full Solution
Consider a function f, such that \(\large f(x) = 5.6\cos(\frac{\pi}{a}(x-1))+b\) , \(\large 0\le x\le 15\), \(\large a,b\in \mathbf{R}\)
The function f has a local maximum at the point (1 , 8.8) and a local minimum at the point (10 , -2.4)
a) Find the period of the function
b) Hence, find the value of a.
c) Find the value of b.
Hint
Full Solution
Consider a function f, such that \(\large f(x) = a\sin(\frac{\pi}{15}(x+2))+b\) , \(\large a,b\in \mathbf{R}\)
The function f has passes through the points (10.5 , 5.5) and (15.5 , 2.5)
Find the value of a and the value of b
Hint
Full Solution
The following diagram shows a ball attached to the end of a spring.
The height, h, in mtres of the ball above the ground t seconds after being released can be modelled by the function
\(\large h(t)=a\cos(\frac{\pi}{b}t)+c\) , \(\large a,b, c\in \mathbf{R}\)
The ball is release from an initial height of 4 metres.
After \(\large \frac{4}{3}\) seconds, the ball is at a height of 1.6 metres.
It takes the ball 4 seconds to return to its initial height.
Find the values of a, b and c.
Hint
Full Solution
How much of Trigonometric Graphs have you understood?
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