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Graphs of trig functions

Module by: Free High School Science Texts Project. E-mail the author

Graphs of Trigonometric Functions

This section describes the graphs of trigonometric functions.

Graph of sinθsinθ

Graph of sinθsinθ

Complete the following table, using your calculator to calculate the values. Then plot the values with sinθsinθ on the yy-axis and θθ on the xx-axis. Round answers to 1 decimal place.

Table 1
θ θ 0 30 60 90 120 150  
sin θ sin θ              
θ θ 180 210 240 270 300 330 360
sin θ sin θ              
 
Figure 1
Figure 1 (MG10C15_016.png)

Let us look back at our values for sinθsinθ

Table 2
θ θ 0 0 30 30 45 45 60 60 90 90 180 180
sin θ sin θ 0 1 2 1 2 1 2 1 2 3 2 3 2 1 0

As you can see, the function sinθsinθ has a value of 0 at θ=0θ=0. Its value then smoothly increases until θ=90θ=90 when its value is 1. We also know that it later decreases to 0 when θ=180θ=180. Putting all this together we can start to picture the full extent of the sine graph. The sine graph is shown in Figure 2. Notice the wave shape, with each wave having a length of 360360. We say the graph has a period of 360360. The height of the wave above (or below) the xx-axis is called the wave's amplitude. Thus the maximum amplitude of the sine-wave is 1, and its minimum amplitude is -1.

Figure 2: The graph of sinθsinθ.
Figure 2 (MG10C15_017.png)

Functions of the form y=asin(x)+qy=asin(x)+q

In the equation, y=asin(x)+qy=asin(x)+q, aa and qq are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 3 for the function f(θ)=2sinθ+3f(θ)=2sinθ+3.

Figure 3: Graph of f(θ)=2sinθ+3f(θ)=2sinθ+3
Figure 3 (trigrep.png)

Functions of the Form y=asin(θ)+qy=asin(θ)+q :

  1. On the same set of axes, plot the following graphs:
    1. a(θ)=sinθ-2a(θ)=sinθ-2
    2. b(θ)=sinθ-1b(θ)=sinθ-1
    3. c(θ)=sinθc(θ)=sinθ
    4. d(θ)=sinθ+1d(θ)=sinθ+1
    5. e(θ)=sinθ+2e(θ)=sinθ+2
    Use your results to deduce the effect of qq.
  2. On the same set of axes, plot the following graphs:
    1. f(θ)=-2·sinθf(θ)=-2·sinθ
    2. g(θ)=-1·sinθg(θ)=-1·sinθ
    3. h(θ)=0·sinθh(θ)=0·sinθ
    4. j(θ)=1·sinθj(θ)=1·sinθ
    5. k(θ)=2·sinθk(θ)=2·sinθ
    Use your results to deduce the effect of aa.

You should have found that the value of aa affects the height of the peaks of the graph. As the magnitude of aa increases, the peaks get higher. As it decreases, the peaks get lower.

qq is called the vertical shift. If q=2q=2, then the whole sine graph shifts up 2 units. If q=-1q=-1, the whole sine graph shifts down 1 unit.

These different properties are summarised in Table 3.

Table 3: Table summarising general shapes and positions of graphs of functions of the form y=asin(x)+qy=asin(x)+q.
  a > 0 a > 0 a < 0 a < 0
q > 0 q > 0
Figure 4
Figure 4 (MG10C15_019.png)
Figure 5
Figure 5 (MG10C15_020.png)
q < 0 q < 0
Figure 6
Figure 6 (MG10C15_021.png)
Figure 7
Figure 7 (MG10C15_022.png)

Domain and Range

For f(θ)=asin(θ)+qf(θ)=asin(θ)+q, the domain is {θ:θR}{θ:θR} because there is no value of θRθR for which f(θ)f(θ) is undefined.

The range of f(θ)=asinθ+qf(θ)=asinθ+q depends on whether the value for aa is positive or negative. We will consider these two cases separately.

If a>0a>0 we have:

- 1 sin θ 1 - a a sin θ a - a + q a sin θ + q a + q - a + q f ( θ ) a + q - 1 sin θ 1 - a a sin θ a - a + q a sin θ + q a + q - a + q f ( θ ) a + q
(1)

This tells us that for all values of θθ, f(θ)f(θ) is always between -a+q-a+q and a+qa+q. Therefore if a>0a>0, the range of f(θ)=asinθ+qf(θ)=asinθ+q is {f(θ):f(θ)[-a+q,a+q]}{f(θ):f(θ)[-a+q,a+q]}.

Similarly, it can be shown that if a<0a<0, the range of f(θ)=asinθ+qf(θ)=asinθ+q is {f(θ):f(θ)[a+q,-a+q]}{f(θ):f(θ)[a+q,-a+q]}. This is left as an exercise.

Tip:
The easiest way to find the range is simply to look for the "bottom" and the "top" of the graph.

Intercepts

The yy-intercept, yintyint, of f(θ)=asin(x)+qf(θ)=asin(x)+q is simply the value of f(θ)f(θ) at θ=0θ=0.

y i n t = f ( 0 ) = a sin ( 0 ) + q = a ( 0 ) + q = q y i n t = f ( 0 ) = a sin ( 0 ) + q = a ( 0 ) + q = q
(2)

Graph of cosθcosθ

Graph of cosθcosθ :

Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with cosθcosθ on the yy-axis and θθ on the xx-axis.

Table 4
θ θ 0 30 60 90 120 150  
cos θ cos θ              
θ θ 180 210 240 270 300 330 360
cos θ cos θ              
 
Figure 8
Figure 8 (MG10C15_023.png)

Let us look back at our values for cosθcosθ

Table 5
θ θ 0 0 30 30 45 45 60 60 90 90 180 180
cos θ cos θ 1 3 2 3 2 1 2 1 2 1 2 1 2 0 - 1 - 1

If you look carefully, you will notice that the cosine of an angle θθ is the same as the sine of the angle 90-θ90-θ. Take for example,

cos 60 = 1 2 = sin 30 = sin ( 90 - 60 ) cos 60 = 1 2 = sin 30 = sin ( 90 - 60 )
(3)

This tells us that in order to create the cosine graph, all we need to do is to shift the sine graph 9090 to the left. The graph of cosθcosθ is shown in Figure 9. As the cosine graph is simply a shifted sine graph, it will have the same period and amplitude as the sine graph.

Figure 9: The graph of cosθcosθ.
Figure 9 (MG10C15_024.png)

Functions of the form y=acos(x)+qy=acos(x)+q

In the equation, y=acos(x)+qy=acos(x)+q, aa and qq are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 10 for the function f(θ)=2cosθ+3f(θ)=2cosθ+3.

Figure 10: Graph of f(θ)=2cosθ+3f(θ)=2cosθ+3
Figure 10 (trigrep1.png)

Functions of the Form y=acos(θ)+qy=acos(θ)+q :

  1. On the same set of axes, plot the following graphs:
    1. a(θ)=cosθ-2a(θ)=cosθ-2
    2. b(θ)=cosθ-1b(θ)=cosθ-1
    3. c(θ)=cosθc(θ)=cosθ
    4. d(θ)=cosθ+1d(θ)=cosθ+1
    5. e(θ)=cosθ+2e(θ)=cosθ+2
    Use your results to deduce the effect of qq.
  2. On the same set of axes, plot the following graphs:
    1. f(θ)=-2·cosθf(θ)=-2·cosθ
    2. g(θ)=-1·cosθg(θ)=-1·cosθ
    3. h(θ)=0·cosθh(θ)=0·cosθ
    4. j(θ)=1·cosθj(θ)=1·cosθ
    5. k(θ)=2·cosθk(θ)=2·cosθ
    Use your results to deduce the effect of aa.

You should have found that the value of aa affects the amplitude of the cosine graph in the same way it did for the sine graph.

You should have also found that the value of qq shifts the cosine graph in the same way as it did the sine graph.

These different properties are summarised in Table 6.

Table 6: Table summarising general shapes and positions of graphs of functions of the form y=acos(x)+qy=acos(x)+q.
  a > 0 a > 0 a < 0 a < 0
q > 0 q > 0
Figure 11
Figure 11 (MG10C15_026.png)
Figure 12
Figure 12 (MG10C15_027.png)
q < 0 q < 0
Figure 13
Figure 13 (MG10C15_028.png)
Figure 14
Figure 14 (MG10C15_029.png)

Domain and Range

For f(θ)=acos(θ)+qf(θ)=acos(θ)+q, the domain is {θ:θR}{θ:θR} because there is no value of θRθR for which f(θ)f(θ) is undefined.

It is easy to see that the range of f(θ)f(θ) will be the same as the range of asin(θ)+qasin(θ)+q. This is because the maximum and minimum values of acos(θ)+qacos(θ)+q will be the same as the maximum and minimum values of asin(θ)+qasin(θ)+q.

Intercepts

The yy-intercept of f(θ)=acos(x)+qf(θ)=acos(x)+q is calculated in the same way as for sine.

y i n t = f ( 0 ) = a cos ( 0 ) + q = a ( 1 ) + q = a + q y i n t = f ( 0 ) = a cos ( 0 ) + q = a ( 1 ) + q = a + q
(4)

Comparison of Graphs of sinθsinθ and cosθcosθ

Figure 15: The graph of cosθcosθ (solid-line) and the graph of sinθsinθ (dashed-line).
Figure 15 (MG10C15_030.png)

Notice that the two graphs look very similar. Both oscillate up and down around the xx-axis as you move along the axis. The distances between the peaks of the two graphs is the same and is constant along each graph. The height of the peaks and the depths of the troughs are the same.

The only difference is that the sinsin graph is shifted a little to the right of the coscos graph by 90. That means that if you shift the whole coscos graph to the right by 90 it will overlap perfectly with the sinsin graph. You could also move the sinsin graph by 90 to the left and it would overlap perfectly with the coscos graph. This means that:

sin θ = cos ( θ - 90 ) ( shift the cos graph to the right ) a nd cos θ = sin ( θ + 90 ) ( shift the sin graph to the left ) sin θ = cos ( θ - 90 ) ( shift the cos graph to the right ) a nd cos θ = sin ( θ + 90 ) ( shift the sin graph to the left )
(5)

Graph of tanθtanθ

Graph of tanθtanθ

Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with tanθtanθ on the yy-axis and θθ on the xx-axis.

Table 7
θ θ 0 30 60 90 120 150  
tan θ tan θ              
θ θ 180 210 240 270 300 330 360
tan θ tan θ              
 
Figure 16
Figure 16 (MG10C15_031.png)

Let us look back at our values for tanθtanθ

Table 8
θ θ 0 0 30 30 45 45 60 60 90 90 180 180
tan θ tan θ 0 1 3 1 3 1 3 3 0

Now that we have graphs for sinθsinθ and cosθcosθ, there is an easy way to visualise the tangent graph. Let us look back at our definitions of sinθsinθ and cosθcosθ for a right-angled triangle.

sin θ cos θ = opposite hypotenuse adjacent hypotenuse = opposite adjacent = tan θ sin θ cos θ = opposite hypotenuse adjacent hypotenuse = opposite adjacent = tan θ
(6)

This is the first of an important set of equations called trigonometric identities. An identity is an equation, which holds true for any value which is put into it. In this case we have shown that

tan θ = sin θ cos θ tan θ = sin θ cos θ
(7)

for any value of θθ.

So we know that for values of θθ for which sinθ=0sinθ=0, we must also have tanθ=0tanθ=0. Also, if cosθ=0cosθ=0 our value of tanθtanθ is undefined as we cannot divide by 0. The graph is shown in Figure 17. The dashed vertical lines are at the values of θθ where tanθtanθ is not defined.

Figure 17: The graph of tanθtanθ.
Figure 17 (MG10C15_044.png)

Functions of the form y=atan(x)+qy=atan(x)+q

In the figure below is an example of a function of the form y=atan(x)+qy=atan(x)+q.

Figure 18: The graph of 2tanθ+12tanθ+1.
Figure 18 (MG10C15_045.png)

Functions of the Form y=atan(θ)+qy=atan(θ)+q :

  1. On the same set of axes, plot the following graphs:
    1. a(θ)=tanθ-2a(θ)=tanθ-2
    2. b(θ)=tanθ-1b(θ)=tanθ-1
    3. c(θ)=tanθc(θ)=tanθ
    4. d(θ)=tanθ+1d(θ)=tanθ+1
    5. e(θ)=tanθ+2e(θ)=tanθ+2
    Use your results to deduce the effect of qq.
  2. On the same set of axes, plot the following graphs:
    1. f(θ)=-2·tanθf(θ)=-2·tanθ
    2. g(θ)=-1·tanθg(θ)=-1·tanθ
    3. h(θ)=0·tanθh(θ)=0·tanθ
    4. j(θ)=1·tanθj(θ)=1·tanθ
    5. k(θ)=2·tanθk(θ)=2·tanθ
    Use your results to deduce the effect of aa.

You should have found that the value of aa affects the steepness of each of the branches. The larger the absolute magnitude of a, the quicker the branches approach their asymptotes, the values where they are not defined. Negative aa values switch the direction of the branches. You should have also found that the value of qq affects the vertical shift as for sinθsinθ and cosθcosθ. These different properties are summarised in Table 9.

Table 9: Table summarising general shapes and positions of graphs of functions of the form y=atan(x)+qy=atan(x)+q.
  a > 0 a > 0 a < 0 a < 0
q > 0 q > 0
Figure 19
Figure 19 (MG10C15_034.png)
Figure 20
Figure 20 (MG10C15_035.png)
q < 0 q < 0
Figure 21
Figure 21 (MG10C15_036.png)
Figure 22
Figure 22 (MG10C15_037.png)

Domain and Range

The domain of f(θ)=atan(θ)+qf(θ)=atan(θ)+q is all the values of θθ such that cosθcosθ is not equal to 0. We have already seen that when cosθ=0cosθ=0, tanθ=sinθcosθtanθ=sinθcosθ is undefined, as we have division by zero. We know that cosθ=0cosθ=0 for all θ=90+180nθ=90+180n, where nn is an integer. So the domain of f(θ)=atan(θ)+qf(θ)=atan(θ)+q is all values of θθ, except the values θ=90+180nθ=90+180n.

The range of f(θ)=atanθ+qf(θ)=atanθ+q is {f(θ):f(θ)(-,)}{f(θ):f(θ)(-,)}.

Intercepts

The yy-intercept, yintyint, of f(θ)=atan(x)+qf(θ)=atan(x)+q is again simply the value of f(θ)f(θ) at θ=0θ=0.

y i n t = f ( 0 ) = a tan ( 0 ) + q = a ( 0 ) + q = q y i n t = f ( 0 ) = a tan ( 0 ) + q = a ( 0 ) + q = q
(8)

Asymptotes

As θθ approaches 9090, tanθtanθ approaches infinity. But as θθ is undefined at 9090, θθ can only approach 9090, but never equal it. Thus the tanθtanθ curve gets closer and closer to the line θ=90θ=90, without ever touching it. Thus the line θ=90θ=90 is an asymptote of tanθtanθ. tanθtanθ also has asymptotes at θ=90+180nθ=90+180n, where nn is an integer.

Graphs of Trigonometric Functions
  1. Using your knowldge of the effects of aa and qq, sketch each of the following graphs, without using a table of values, for θ[0;360]θ[0;360]
    1. y=2sinθy=2sinθ
    2. y=-4cosθy=-4cosθ
    3. y=-2cosθ+1y=-2cosθ+1
    4. y=sinθ-3y=sinθ-3
    5. y=tanθ-2y=tanθ-2
    6. y=2cosθ-1y=2cosθ-1
    Click here for the solution.
  2. Give the equations of each of the following graphs:
    Figure 23
    Figure 23 (trigrep4.png)
    Figure 24
    Figure 24 (trigrep5.png)
    Figure 25
    Figure 25 (trigrep6.png)
    Click here for the solution.

The following presentation summarises what you have learnt in this chapter.

Figure 26

Summary

  • We can define three trigonometric functions for right angled triangles: sine (sin), cosine (cos) and tangent (tan).
  • Each of these functions have a reciprocal: cosecant (cosec), secant (sec) and cotangent (cot).
  • We can use the principles of solving equations and the trigonometric functions to help us solve simple trigonometric equations.
  • We can solve problems in two dimensions that involve right angled triangles.
  • For some special angles, we can easily find the values of sin, cos and tan.
  • We can extend the definitions of the trigonometric functions to any angle.
  • Trigonometry is used to help us solve problems in 2-dimensions, such as finding the height of a building.
  • We can draw graphs for sin, cos and tan

End of Chapter Exercises

  1. Calculate the unknown lengths
    Figure 27
    Figure 27 (MG10C15_041.png)
    Click here for the solution.
  2. In the triangle PQRPQR, PR=20PR=20 cm, QR=22QR=22 cm and PR^Q=30PR^Q=30. The perpendicular line from PP to QRQR intersects QRQR at XX. Calculate
    1. the length XRXR,
    2. the length PXPX, and
    3. the angle QP^XQP^X
    Click here for the solution.
  3. A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder?
    Click here for the solution.
  4. A ladder of length 25 m is resting against a wall, the ladder makes an angle 3737 to the wall. Find the distance between the wall and the base of the ladder?
    Click here for the solution.
  5. In the following triangle find the angle AB^CAB^C
    Figure 28
    Figure 28 (MG10C15_042.png)
    Click here for the solution.
  6. In the following triangle find the length of side CDCD
    Figure 29
    Figure 29 (MG10C15_043.png)
    Click here for the solution.
  7. A(5;0)A(5;0) and B(11;4)B(11;4). Find the angle between the line through A and B and the x-axis.
    Click here for the solution.
  8. C(0;-13)C(0;-13) and D(-12;14)D(-12;14). Find the angle between the line through C and D and the y-axis.
    Click here for the solution.
  9. A 5m5m ladder is placed 2m2m from the wall. What is the angle the ladder makes with the wall?
    Click here for the solution.
  10. Given the points: E(5;0), F(6;2) and G(8;-2), find angle FE^GFE^G.
    Click here for the solution.
  11. An isosceles triangle has sides 9 cm ,9 cm 9 cm ,9 cm and 2 cm 2 cm . Find the size of the smallest angle of the triangle.
    Click here for the solution.
  12. A right-angled triangle has hypotenuse 13 mm 13 mm . Find the length of the other two sides if one of the angles of the triangle is 5050.
    Click here for the solution.
  13. One of the angles of a rhombus (rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter 20 cm 20 cm is 3030.
    1. Find the sides of the rhombus.
    2. Find the length of both diagonals.
    Click here for the solution.
  14. Captain Hook was sailing towards a lighthouse with a height of 10m10m.
    1. If the top of the lighthouse is 30m30m away, what is the angle of elevation of the boat to the nearest integer?
    2. If the boat moves another 7m7m towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
    Click here for the solution.
  15. (Tricky) A triangle with angles 40,4040,40 and 100100 has a perimeter of 20 cm 20 cm . Find the length of each side of the triangle.
    Click here for the solution.

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