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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Graphs of the sine, cosine and tangent functions

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Graph of sine and cosine functions
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Factorisation amd defactorisation of trigonometric functions
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Graph of tangent and cotangent functions
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Applications of the derivative
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Trigonometric equations
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↓   Show theory [5]   ↓


Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:graphs_of_the_sine%2C_cosine_and_tangent_functions/5409/


Example 1.
Draw the graph of the function and write down the range of values:

    
Example 2.

In the same coordinate system draw a graph of the function ; and line y = 1 and write the coordinates of the intersections.

    
3.

The function is given.


  • Calculate accurately


  • Write down the zeros and intercepts of the function and draw its graph

    
4.

Show that the period of the function is and draw its graph.

    
5.
Draw the graph of the function and write down the range of values:

    
6.
Draw a graph of the function:

    
7.
Draw the graph of the function and write down the range of values:

    
8.
Draw the graph of the function and write down the range of values:

    

9.
Draw a graph of the function:

    
10.
Draw the graph of the function and write down the range of values:

    
11.

Draw a graph of the functions and . Also record the period and range of the values of the functions and .

    
12.

Choose the correct answer in each task:


  • Find the value of .


     

  • Simplify .


     

  • The derivative of the function .


     

  • Simplify , .


     
    
13.
Draw the graph of the function and write down the range of values:

    
14.
Draw the graph of the function and write down the range of values:

    
15.
Write down the basic period and amplitude of the function:

    
16.
Write down the basic period and amplitude of the function:

    
17.
Draw the following graphs in the same coordinate system:

    
18.
Write down the basic period and amplitude of the function:

    

19.

Given the graph of the function , first move it parallel by in the direction of the x-axis, reflect over the x-axis, then stretch it by a factor of - 2 in the direction of the y-axis and then move it in parallel by 1 in the direction y-axis.


  • Obtain the equation of the obtained function.


  • Obtain the domain and range of the function.


  • Calculate the zeros of the function.

    
20.

Draw the graphs of the following in the same coordinate system:


    
21.
Draw the following graphs in the same coordinate system:

    
22.

Draw the graphs of the following in the same coordinate system:


    
23.
Draw the graph of the function:

    
24.

In the same coordinate system, plot the graph of the functions f and g.




where






Write down their intersections.

    
25.
Draw the graph of the function:

    
26.

The function is given.


  • Draw the graph of the function on the interval .


  • Obtain the zeros of the function and its range of values.


  • At what interval is the function decreasing?

    
27.

The function is given.


  • Obtain the zeros of the function.


  • Draw the graph of the function f and write down its range of values.


  • Obtain the abscissas of the maxima and minima of the function.


  • Draw the graph of the function in the same coordinate system

    
28.

In the same coordinate system draw the graph of functions .




and




and write down their intersections.

    

29.

Draw the graph of the functions below in the same coordinate system:




and




and write down their intersections.

    
30.
Draw the following graphs in the same coordinate system:

    
31.
Write down the zeros and abscissas of the maxima and minima of a given function.

    
32.
Write down the zeros and abscissas of the maxima and minima of a given function.

    
33.
Write down the zeros and abscissas of the maxima and minima of a given function.

    
34.

Draw graphs of the functions and in the same coordinate system and calculate the abscissas of the intersections of their graphs.

    
35.
Write down the zeros and abscissas of the maxima and minima of a given function.

    
36.
Solve the equation:

    
37.
Draw the graph of the function below and obtain the range of values:

    
38.
Draw the graph of:

    

39.

The function is given.


  • Calculate the zeros of the function


  • Write the function as


  • Draw a graph of the function and write down the stock of values

    
40.
Draw the graph of:

    
41.

Draw graphs and in the same coordinate system and calculate the abscissas of their intersections.

    
42.
Draw the graph of the function below and obtain the range of values:

    
43.
Draw the graph of the function below and obtain the range of values:

    
44.
Draw the graph of:

    
45.

Fill in the table:


Function    
Amplitude A   
k   
Period  
Range    
    
46.

Given the function f:


  • Draw the graph of the function f.


  • Read from the graph where .


  • Draw the graph of the function g:

   
47.

Draw the graph of the function:


    
48.

The graph below shows the changes in the water depth h [t] in one day. The time t is given in hours h, from midnight to noon.




  • From the graph, read when the water reaches a minimum.


  • From the graph, read when the water reaches its maximum.


  • The water depth is represented by the function


    • Show that A = 8


    • Calculate the value C.


    • Calculate the value B.


    • A sailor knows he cannot sail if the water depth is less than 12 m. Calculate the time when the sailor cannot sail.

    

49.

A wave is described by the function , where:


  • is the current offset of the observed point


  • is the wave amplitude of the oscillating particles of matter


  • is the wave frequency


  • is the wave velocity


  • the distance of the observed point from the source of the wave


  • is the elapsed time


The wave propagates through a substance, which is described by the equation:




The sketch below is for the performance only, it does not represent the given function:




  • Obtain the wave frequency and the wave speed .


  • What is the distance from the wave source to the point that is 36cm farther away after 10 seconds?


  • What is the displacement of the point located 14cm farther away from the source at the same instant?


    
50.

A patient's blood pressure can be expressed as:




where is the blood pressure measured at time t.


[Pressure is measured in mmHg (millimeter of mercury) and is a unit for measuring blood pressure in medicine; one millimeter of mercury corresponds to 133.322 pascals.]


  • Determine the amplitude and period of the function p.


  • Determine the maximum blood pressure that can be measured and calculate when it occurs.


  • Sketch the graph p (t).


  • Find the first time the device detects 110 mmHg.

    
51.

The function is given.


  • Draw the graph of the function and write down the zeros of the function.


  • Read from the graph where .


  • Obtain the abscissas of the maxima and minima of the function.


  • Draw the graph of the the function in the same coordinate system

    
52.

In the same coordinate system, draw the graph of the functions f and g.




where






Obtain their points of intersections.

    
53.

In the interval draw graphs of functions below and express them in the form .









    
54.
Draw the graph of:

    
55.

Draw the graph of the functions below, obtain the range of values and calculate the zeros.





    
56.

Obtain the range, domain, base period, and the function f whose graph is drawn below.


    
57.

The function h: , is given.


  • Obtain the abscissas of the maxima and minima of the function and its zeros.


  • Draw the graph of the function h.


  • At what interval is the function increasing?

    
58.

Given the function . Calculate:


  • The zeros.


  • The abscissa of the maximum and minimum points.


  • Draw the graph of the function.

    

59.

For what values of x does the function equal ?

    
60.

The function is given.


Draw the graph of the function and solve the inequality .

    
61.

On the interval , draw the graph of the function and write the union of the intervals on which the function takes non-negative values.

    
62.

Obtain the rule for the function f given below. What is the basic period of the function f?


    
63.

The population of rabbits , measured in thousands, in the northern part of South Australia is roughly determined by the equation:






t is the number of months measured after 1st. of January.


  • Sketch the graph of .


  • What can be the largest population of rabbits?


  • How much time between two maximum rabbit populations?

    
64.

The function is given. Calculate the zeros of this function and draw its graph in the given coordinate system.


    
65.

Given the function , obtain the intersection of the function f with the ordinate axis and calculate the zeros of this function.

    
66.

Calculate the zeros of the function and draw its graph.


    
67.

The functions and are given.


  • Draw the graph of the functions on the interval . Calculate the intersection of these graphs on .

  • Calculate the angle at which the graphs of the function and intersect on the interval .

  • Calculate the area of the figure enclosed by the graphs of the functions and on the interval .

  • For which real numbers does the curves:




    and




    intersect at interval ?

    
68.

Given the function:




  • Determine the range of values, zeros and abscissa of minima and maxima of the function.

  • Draw the graph of the function on the interval .

  • Calculate the exact value of .

    

69.

Draw the graph of the function on the interval . Obtain the domain and range of values of the function .

    
70.

Draw the graph of the function on the interval . Also, obtain the domain and range of values of the function.

    
71.

Solution:


Given the graph below on the interval ,




read the:


  • zeros of the functions,

  • range of values,

  • value of the variable x for which the value of the function is 1.

    
72.

Given the function .


  • Draw the graph of the function.

  • Calculate the zeros, intersect on the y-axis and the range of values of the function .

  • Draw the graph of the function .