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Trigonometry: Graphs of trig functions (Grade 11)

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

History of Trigonometry

Work in pairs or groups and investigate the history of the development of trigonometry. Describe the various stages of development and how different cultures used trigonometry to improve their lives.

The works of the following people or cultures can be investigated:

  1. Cultures
    1. Ancient Egyptians
    2. Mesopotamians
    3. Ancient Indians of the Indus Valley
  2. People
    1. Lagadha (circa 1350-1200 BC)
    2. Hipparchus (circa 150 BC)
    3. Ptolemy (circa 100)
    4. Aryabhata (circa 499)
    5. Omar Khayyam (1048-1131)
    6. Bhaskara (circa 1150)
    7. Nasir al-Din (13th century)
    8. al-Kashi and Ulugh Beg (14th century)
    9. Bartholemaeus Pitiscus (1595)

Graphs of Trigonometric Functions

Functions of the form y=sin(kθ)y=sin(kθ)

In the equation, y=sin(kθ)y=sin(kθ), kk is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 1 for the function f(θ)=sin(2θ)f(θ)=sin(2θ).

Figure 1: Graph of f(θ)=sin(2θ)f(θ)=sin(2θ) (solid line) and the graph of g(θ)=sin(θ)g(θ)=sin(θ) (dotted line).
Figure 1 (MG11C17_001.png)

Functions of the form y=sin(kθ)y=sin(kθ)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = sin 0 , 5 θ a ( θ ) = sin 0 , 5 θ
  2. b ( θ ) = sin 1 θ b ( θ ) = sin 1 θ
  3. c ( θ ) = sin 1 , 5 θ c ( θ ) = sin 1 , 5 θ
  4. d ( θ ) = sin 2 θ d ( θ ) = sin 2 θ
  5. e ( θ ) = sin 2 , 5 θ e ( θ ) = sin 2 , 5 θ

Use your results to deduce the effect of kk.

You should have found that the value of kk affects the period or frequency of the graph. Notice that in the case of the sine graph, the period (length of one wave) is given by 360∘k360∘k.

These different properties are summarised in Table 1.

Table 1: Table summarising general shapes and positions of graphs of functions of the form y=sin(kx)y=sin(kx). The curve y=sin(x)y=sin(x) is shown as a dotted line.
k > 0 k > 0 k < 0 k < 0
Figure 2
Figure 2 (MG11C17_002.png)
Figure 3
Figure 3 (MG11C17_003.png)

Domain and Range

For f(θ)=sin(kθ)f(θ)=sin(kθ), the domain is {θ:θ∈R}{θ:θ∈R} because there is no value of θ∈Rθ∈R for which f(θ)f(θ) is undefined.

The range of f(θ)=sin(kθ)f(θ)=sin(kθ) is {f(θ):f(θ)∈[-1,1]}{f(θ):f(θ)∈[-1,1]}.

Intercepts

For functions of the form, y=sin(kθ)y=sin(kθ), the details of calculating the intercepts with the yy axis are given.

There are many xx-intercepts.

The yy-intercept is calculated by setting θ=0θ=0:

y = sin ( k θ ) y i n t = sin ( 0 ) = 0 y = sin ( k θ ) y i n t = sin ( 0 ) = 0
(1)

Functions of the form y=cos(kθ)y=cos(kθ)

In the equation, y=cos(kθ)y=cos(kθ), kk is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 4 for the function f(θ)=cos(2θ)f(θ)=cos(2θ).

Figure 4: Graph of f(θ)=cos(2θ)f(θ)=cos(2θ) (solid line) and the graph of g(θ)=cos(θ)g(θ)=cos(θ) (dotted line).
Figure 4 (MG11C17_004.png)

Functions of the form y=cos(kθ)y=cos(kθ)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = cos 0 , 5 θ a ( θ ) = cos 0 , 5 θ
  2. b ( θ ) = cos 1 θ b ( θ ) = cos 1 θ
  3. c ( θ ) = cos 1 , 5 θ c ( θ ) = cos 1 , 5 θ
  4. d ( θ ) = cos 2 θ d ( θ ) = cos 2 θ
  5. e ( θ ) = cos 2 , 5 θ e ( θ ) = cos 2 , 5 θ

Use your results to deduce the effect of kk.

You should have found that the value of kk affects the period or frequency of the graph. The period of the cosine graph is given by 360∘k360∘k.

These different properties are summarised in Table 2.

Table 2: Table summarising general shapes and positions of graphs of functions of the form y=cos(kx)y=cos(kx). The curve y=cos(x)y=cos(x) is plotted with a dotted line.
k > 0 k > 0 k < 0 k < 0
Figure 5
Figure 5 (MG11C17_005.png)
Figure 6
Figure 6 (MG11C17_006.png)

Domain and Range

For f(θ)=cos(kθ)f(θ)=cos(kθ), the domain is {θ:θ∈R}{θ:θ∈R} because there is no value of θ∈Rθ∈R for which f(θ)f(θ) is undefined.

The range of f(θ)=cos(kθ)f(θ)=cos(kθ) is {f(θ):f(θ)∈[-1,1]}{f(θ):f(θ)∈[-1,1]}.

Intercepts

For functions of the form, y=cos(kθ)y=cos(kθ), the details of calculating the intercepts with the yy axis are given.

The yy-intercept is calculated as follows:

y = cos ( k θ ) y i n t = cos ( 0 ) = 1 y = cos ( k θ ) y i n t = cos ( 0 ) = 1
(2)

Functions of the form y=tan(kθ)y=tan(kθ)

In the equation, y=tan(kθ)y=tan(kθ), kk is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 7 for the function f(θ)=tan(2θ)f(θ)=tan(2θ).

Figure 7: The graph of tan(2θ)tan(2θ) (solid line) and the graph of g(θ)=tan(θ)g(θ)=tan(θ) (dotted line). The asymptotes are shown as dashed lines.
Figure 7 (MG11C17_007.png)

Functions of the form y=tan(kθ)y=tan(kθ)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = tan 0 , 5 θ a ( θ ) = tan 0 , 5 θ
  2. b ( θ ) = tan 1 θ b ( θ ) = tan 1 θ
  3. c ( θ ) = tan 1 , 5 θ c ( θ ) = tan 1 , 5 θ
  4. d ( θ ) = tan 2 θ d ( θ ) = tan 2 θ
  5. e ( θ ) = tan 2 , 5 θ e ( θ ) = tan 2 , 5 θ

Use your results to deduce the effect of kk.

You should have found that, once again, the value of kk affects the periodicity (i.e. frequency) of the graph. As kk increases, the graph is more tightly packed. As kk decreases, the graph is more spread out. The period of the tan graph is given by 180∘k180∘k.

These different properties are summarised in Table 3.

Table 3: Table summarising general shapes and positions of graphs of functions of the form y=tan(kθ)y=tan(kθ).
k > 0 k > 0 k < 0 k < 0
Figure 8
Figure 8 (MG11C17_008.png)
Figure 9
Figure 9 (MG11C17_009.png)

Domain and Range

For f(θ)=tan(kθ)f(θ)=tan(kθ), the domain of one branch is {θ:θ∈(-90∘k,90∘k)}{θ:θ∈(-90∘k,90∘k)} because the function is undefined for θ=-90∘kθ=-90∘k and θ=90∘kθ=90∘k.

The range of f(θ)=tan(kθ)f(θ)=tan(kθ) is {f(θ):f(θ)∈(-∞,∞)}{f(θ):f(θ)∈(-∞,∞)}.

Intercepts

For functions of the form, y=tan(kθ)y=tan(kθ), the details of calculating the intercepts with the xx and yy axis are given.

There are many xx-intercepts; each one is halfway between the asymptotes.

The yy-intercept is calculated as follows:

y = tan ( k θ ) y i n t = tan ( 0 ) = 0 y = tan ( k θ ) y i n t = tan ( 0 ) = 0
(3)

Asymptotes

The graph of tankθtankθ has asymptotes because as kθkθ approaches 90∘90∘, tankθtankθ approaches infinity. In other words, there is no defined value of the function at the asymptote values.

Functions of the form y=sin(θ+p)y=sin(θ+p)

In the equation, y=sin(θ+p)y=sin(θ+p), pp is a constant and has 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)=sin(θ+30∘)f(θ)=sin(θ+30∘).

Figure 10: Graph of f(θ)=sin(θ+30∘)f(θ)=sin(θ+30∘) (solid line) and the graph of g(θ)=sin(θ)g(θ)=sin(θ) (dotted line).
Figure 10 (MG11C17_010.png)

Functions of the Form y=sin(θ+p)y=sin(θ+p)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = sin ( θ - 90 ∘ ) a ( θ ) = sin ( θ - 90 ∘ )
  2. b ( θ ) = sin ( θ - 60 ∘ ) b ( θ ) = sin ( θ - 60 ∘ )
  3. c ( θ ) = sin θ c ( θ ) = sin θ
  4. d ( θ ) = sin ( θ + 90 ∘ ) d ( θ ) = sin ( θ + 90 ∘ )
  5. e ( θ ) = sin ( θ + 180 ∘ ) e ( θ ) = sin ( θ + 180 ∘ )

Use your results to deduce the effect of pp.

You should have found that the value of pp affects the position of the graph along the yy-axis (i.e. the yy-intercept) and the position of the graph along the xx-axis (i.e. the phase shift). The pp value shifts the graph horizontally. If pp is positive, the graph shifts left and if pp is negative tha graph shifts right.

These different properties are summarised in Table 4.

Table 4: Table summarising general shapes and positions of graphs of functions of the form y=sin(θ+p)y=sin(θ+p). The curve y=sin(θ)y=sin(θ) is plotted with a dotted line.
p > 0 p > 0 p < 0 p < 0
Figure 11
Figure 11 (MG11C17_011.png)
Figure 12
Figure 12 (MG11C17_012.png)

Domain and Range

For f(θ)=sin(θ+p)f(θ)=sin(θ+p), the domain is {θ:θ∈R}{θ:θ∈R} because there is no value of θ∈Rθ∈R for which f(θ)f(θ) is undefined.

The range of f(θ)=sin(θ+p)f(θ)=sin(θ+p) is {f(θ):f(θ)∈[-1,1]}{f(θ):f(θ)∈[-1,1]}.

Intercepts

For functions of the form, y=sin(θ+p)y=sin(θ+p), the details of calculating the intercept with the yy axis are given.

The yy-intercept is calculated as follows: set θ=0∘θ=0∘

y = sin ( θ + p ) y i n t = sin ( 0 + p ) = sin ( p ) y = sin ( θ + p ) y i n t = sin ( 0 + p ) = sin ( p )
(4)

Functions of the form y=cos(θ+p)y=cos(θ+p)

In the equation, y=cos(θ+p)y=cos(θ+p), pp is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 13 for the function f(θ)=cos(θ+30∘)f(θ)=cos(θ+30∘).

Figure 13: Graph of f(θ)=cos(θ+30∘)f(θ)=cos(θ+30∘) (solid line) and the graph of g(θ)=cos(θ)g(θ)=cos(θ) (dotted line).
Figure 13 (MG11C17_013.png)

Functions of the Form y=cos(θ+p)y=cos(θ+p)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = cos ( θ - 90 ∘ ) a ( θ ) = cos ( θ - 90 ∘ )
  2. b ( θ ) = cos ( θ - 60 ∘ ) b ( θ ) = cos ( θ - 60 ∘ )
  3. c ( θ ) = cos θ c ( θ ) = cos θ
  4. d ( θ ) = cos ( θ + 90 ∘ ) d ( θ ) = cos ( θ + 90 ∘ )
  5. e ( θ ) = cos ( θ + 180 ∘ ) e ( θ ) = cos ( θ + 180 ∘ )

Use your results to deduce the effect of pp.

You should have found that the value of pp affects the yy-intercept and phase shift of the graph. As in the case of the sine graph, positive values of pp shift the cosine graph left while negative pp values shift the graph right.

These different properties are summarised in Table 5.

Table 5: Table summarising general shapes and positions of graphs of functions of the form y=cos(θ+p)y=cos(θ+p). The curve y=cosθy=cosθ is plotted with a dotted line.
p > 0 p > 0 p < 0 p < 0
Figure 14
Figure 14 (MG11C17_014.png)
Figure 15
Figure 15 (MG11C17_015.png)

Domain and Range

For f(θ)=cos(θ+p)f(θ)=cos(θ+p), the domain is {θ:θ∈R}{θ:θ∈R} because there is no value of θ∈Rθ∈R for which f(θ)f(θ) is undefined.

The range of f(θ)=cos(θ+p)f(θ)=cos(θ+p) is {f(θ):f(θ)∈[-1,1]}{f(θ):f(θ)∈[-1,1]}.

Intercepts

For functions of the form, y=cos(θ+p)y=cos(θ+p), the details of calculating the intercept with the yy axis are given.

The yy-intercept is calculated as follows: set θ=0∘θ=0∘

y = cos ( θ + p ) y i n t = cos ( 0 + p ) = cos ( p ) y = cos ( θ + p ) y i n t = cos ( 0 + p ) = cos ( p )
(5)

Functions of the form y=tan(θ+p)y=tan(θ+p)

In the equation, y=tan(θ+p)y=tan(θ+p), pp is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 16 for the function f(θ)=tan(θ+30∘)f(θ)=tan(θ+30∘).

Figure 16: The graph of tan(θ+30∘)tan(θ+30∘) (solid lines) and the graph of g(θ)=tan(θ)g(θ)=tan(θ) (dotted lines).
Figure 16 (MG11C17_016.png)

Functions of the Form y=tan(θ+p)y=tan(θ+p)

On the same set of axes, plot the following graphs:

  1. a ( θ ) = tan ( θ - 90 ∘ ) a ( θ ) = tan ( θ - 90 ∘ )
  2. b ( θ ) = tan ( θ - 60 ∘ ) b ( θ ) = tan ( θ - 60 ∘ )
  3. c ( θ ) = tan θ c ( θ ) = tan θ
  4. d ( θ ) = tan ( θ + 60 ∘ ) d ( θ ) = tan ( θ + 60 ∘ )
  5. e ( θ ) = tan ( θ + 180 ∘ ) e ( θ ) = tan ( θ + 180 ∘ )

Use your results to deduce the effect of pp.

You should have found that the value of pp once again affects the yy-intercept and phase shift of the graph. There is a horizontal shift to the left if pp is positive and to the right if pp is negative.

These different properties are summarised in Table 6.

Table 6: Table summarising general shapes and positions of graphs of functions of the form y=tan(θ+p)y=tan(θ+p). The curve y=tan(θ)y=tan(θ) is plotted with a dotted line.
k > 0 k > 0 k < 0 k < 0
Figure 17
Figure 17 (MG11C17_017.png)
Figure 18
Figure 18 (MG11C17_018.png)

Domain and Range

For f(θ)=tan(θ+p)f(θ)=tan(θ+p), the domain for one branch is {θ:θ∈(-90∘-p,90∘-p}{θ:θ∈(-90∘-p,90∘-p} because the function is undefined for θ=-90∘-pθ=-90∘-p and θ=90∘-pθ=90∘-p.

The range of f(θ)=tan(θ+p)f(θ)=tan(θ+p) is {f(θ):f(θ)∈(-∞,∞)}{f(θ):f(θ)∈(-∞,∞)}.

Intercepts

For functions of the form, y=tan(θ+p)y=tan(θ+p), the details of calculating the intercepts with the yy axis are given.

The yy-intercept is calculated as follows: set θ=0∘θ=0∘

y = tan ( θ + p ) y i n t = tan ( p ) y = tan ( θ + p ) y i n t = tan ( p )
(6)

Asymptotes

The graph of tan(θ+p)tan(θ+p) has asymptotes because as θ+pθ+p approaches 90∘90∘, tan(θ+p)tan(θ+p) approaches infinity. Thus, there is no defined value of the function at the asymptote values.

Functions of various form

Using your knowledge of the effects of pp and kk draw a rough sketch of the following graphs without a table of values.

  1. y = sin 3 x y = sin 3 x
  2. y = - cos 2 x y = - cos 2 x
  3. y = tan 1 2 x y = tan 1 2 x
  4. y = sin ( x - 45 ∘ ) y = sin ( x - 45 ∘ )
  5. y = cos ( x + 45 ∘ ) y = cos ( x + 45 ∘ )
  6. y = tan ( x - 45 ∘ ) y = tan ( x - 45 ∘ )
  7. y = 2 sin 2 x y = 2 sin 2 x
  8. y = sin ( x + 30 ∘ ) + 1 y = sin ( x + 30 ∘ ) + 1

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