Trigonometric Graphs

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.


Key Concepts

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

Summary

Test Yourself

Here is a quiz about the period of a trigonometric function



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Here is a quiz about the amplitude of a trigonometric function



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Here is a quiz about translations of a trigonometric function



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Exam-style Questions

Question 1

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

 

 

Question 2

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

 

Question 3

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

Question 4

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

Question 5

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

MY PROGRESS

How much of Trigonometric Graphs have you understood?