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Trigonometric functions
By using the addition theorems and other identities also implied by these we can represent and characterise some more, seemingly more compound functions. We selected some of these as examples.
Example 1
Let us plot the graph of the function \latex{f:\R\rightarrow \R,f\left(x\right) =\sin x\times \cos x} and let us characterise the function.
Solution
Let us first transform the formula appearing in the definition of the function by applying the identity relating to the sine of the double-angle:
\latex{f\left(x\right) =\frac{1}{2}\times \sin 2x}.
So the graph of the function \latex{ f } can be derived from the graph of the sine function by shrinking by half along the \latex{ x }-axis and perpendicularly to the \latex{ y }-axis, and then by shrinking the graph obtained by half perpendicularly to the \latex{ x }-axis (Figure 16).
\latex{\frac{\pi }{4} }
\latex{y=\sin2x}
\latex{y=\frac{1 }{2}\times \sin 2x }
\latex{\frac{\pi }{2} }
\latex{\frac{3\pi }{4} }
\latex{\pi}
\latex{-\frac{\pi }{4} }
\latex{-\frac{\pi }{2} }
\latex{-\pi}
\latex{-\frac{3\pi }{4} }
\latex{\frac{1}{2} }
\latex{-\frac{1}{2} }
\latex{1}
\latex{-1}
\latex{y=\sin x}
\latex{ y }
\latex{ x }
Figure 16
The graph of the function \latex{ f } is very “similar” to the graph of the sine function; it is only shrunk by half with respect to the origin.
The function f is periodic with the period \latex{\pi}:
\latex{f\left(x+\pi \right) =\frac{1}{2}\times \sin 2\left(x+\pi \right)=\frac{1}{2}\times \sin \left(2x+2\pi \right)=\frac{1}{2}\times \sin 2x}.
The zeros of \latex{ f } are the numbers \latex{k\times \frac{\pi }{2} \left(k\in \Z\right)}. Its range is the closed interval \latex{\left[-\frac{1}{2};\frac{1}{2} \right]}. On the interval \latex{\left[-\frac{\pi}{4};\frac{\pi}{4} \right]} it is strictly increasing, on the interval \latex{\left[\frac{\pi}{4};3\times\frac{\pi}{4} \right]} it is strictly decreasing.
Its maximums are at the places \latex{\frac{\pi }{4} +k\pi\left(k\in \Z\right)}, the value of f is here is \latex{\frac{1}{2}}. Its minimums are at the places \latex{-\frac{\pi }{4} +n\pi\left(k\in \Z\right)}, the value of f here is \latex{-\frac{1}{2}}.
Example 2
Let us plot the graph of the function \latex{g:\R\rightarrow \R,g\left(x\right) =\sin x+\cos x} and let us characterise the function.
Solution
The following idea might help with plotting the graph.
Let us factor out \latex{\sqrt{2} } from the formula defining the function, and let us apply the addition theorem of the sine function.
\latex{g\left(x\right)=\sqrt{2}\times \left\lgroup\frac{1}{\sqrt{2} }\times \sin x+\frac{1}{\sqrt{2}}\times \cos x \right\rgroup=}
\latex{=\sqrt{2} \times \left\lgroup\sin x\times \cos \frac{\pi }{4}+\cos x\times \sin \frac{\pi }{4} \right\rgroup=\sqrt{2}\times \left\lgroup\sin x+\frac{\pi }{4} \right\rgroup.}
\latex{=\sqrt{2} \times \left\lgroup\sin x\times \cos \frac{\pi }{4}+\cos x\times \sin \frac{\pi }{4} \right\rgroup=\sqrt{2}\times \left\lgroup\sin x+\frac{\pi }{4} \right\rgroup.}
he result shows that \latex{ g } is also a transformation of the sine function.
Its graph is derived from the graph of the sine function by translating it by \latex{\frac{\pi}{4}} to the negative direction along the \latex{ x }-axis, and then by stretching it along the y-axis by \latex{\sqrt{2}} (Figure 17).
\latex{-\pi}
\latex{y=\sin x}
\latex{\pi}
\latex{-\frac{3\pi }{4} }
\latex{-\frac{\pi }{2} }
\latex{-\frac{\pi }{4} }
\latex{\frac{\pi }{4} }
\latex{\frac{\pi }{2} }
\latex{\frac{3\pi }{4} }
\latex{1}
\latex{-1}
\latex{\sqrt{2} }
\latex{-\sqrt{2} }
\latex{y=\sqrt{2} \times \sin \left\lgroup x+\frac{\pi }{4} \right\rgroup }
\latex{y=\sin \left\lgroup x+\frac{\pi }{4} \right\rgroup }
\latex{ y }
\latex{ x }
Figure 17
The function \latex{ g } is also periodic with the period \latex{2\pi}
Its zeros are: \latex{3\times \frac{\pi }{4}+k\pi \left(k\in \Z\right)}.
Its range is the closed interval \latex{\left[-\sqrt{2};\sqrt{2} \right]}.
On the interval \latex{\left[-3\times \frac{\pi }{4};\frac{\pi }{4} \right]} it is strictly increasing.
On the interval \latex{\left[\frac{\pi }{4};5\times \frac{\pi }{4} \right]} it is strictly decreasing.
Its maximums are at the place \latex{\frac{\pi }{4} +2k\pi \left(k\in \Z\right) }.
Its minimums are at the place \latex{5\times\frac{\pi }{4} +2k\pi \left(k\in \Z\right) }.
Example 3
Let us plot the graph of the function \latex{h:\R\rightarrow \R,h\left(x\right) =\sqrt{3}\times \sin x-\cos x}.
Solution
The method applied during the solution of the previous example is effective here too. Let us factor out \latex{ 2 } from the formula giving the value of the function, and let us apply the addition theorem:
\latex{h\left(x\right) =2\times \left\lgroup\frac{\sqrt{3} }{2}\times \sin x-\frac{1}{2}\times \cos x \right\rgroup =2\times \sin \left\lgroup x-\frac{\pi }{6} \right\rgroup}.
So the function \latex{ h } is a transformation of the sine function. Its graph can be derived from the graph of the sine function by translating it by \latex{\frac{\pi }{6}} to the positive direction along the \latex{ x }-axis, and then by stretching it to the direction of the \latex{ y }-axis by a factor of \latex{ 2 } (Figure 18).
\latex{ 1 }
\latex{ 2 }
\latex{ -1 }
\latex{ -2 }
\latex{ y }
\latex{ x }
Figure 18
It is also worth formulating the method used in the previous two examples in general. Let us assume that \latex{ a } and \latex{ b } are two real numbers so that they are not \latex{ 0 } at once, i.e. \latex{a^2+b^2\neq0}. In this case the function given in the following form:
\latex{f:\R\rightarrow \R,f\left(x\right) =a\times \sin x+b\times \cos x},
is a transformation of the sine function. It can be shown as follows.
Let us factor out \latex{\sqrt{a^2+b^2}} from the formula defining f, which is not 0:
\latex{f\left(x\right) =\sqrt{a^2+b^2}\times \left\lgroup\frac{a}{\sqrt{a^2+b^2}}\times \sin x+\frac{b}{\sqrt{a^2+b^2}}\times \cos x \right\rgroup}.
As \latex{\left\lgroup\frac{a}{\sqrt{a^2+b^2} } \right\rgroup^2 +\left\lgroup\frac{b}{\sqrt{a^2+b^2} } \right\rgroup^2=1}, the coefficients of \latex{\sin x} and \latex{\cos x} are the two coordinates of a unit vector. However in this case there is a real number \latex{\alpha} (an angle with the radian measure \latex{\alpha}) for which
\latex{\cos \alpha =\frac{a}{\sqrt{a^2+b^2} }} and \latex{\sin \alpha =\frac{b}{\sqrt{a^2+b^2} }}.
Thus the addition theorem can be applied, and
\latex{f\left(x\right) =\sqrt{a^2+b^2} \times \sin \left(x+\alpha \right)}.
Example 4
Let us plot the graph of the function \latex{f:\R\rightarrow \R,f\left(x\right) =\sin^4x-\cos^4x}.
Solution
By applying the identities let us transform the expression defining the value of the function \latex{ f }:
\latex{f\left(x\right) =\left(\sin^2x+\cos ^2x\right) \times \left(\sin ^2x-\cos ^2x\right) =-\cos 2x}.
The graph of \latex{ f } can be derived from the graph of the cosine function by shrinking it perpendicularly to the \latex{ y }-axis by half, and then by reflecting it about the \latex{ x }-axis (Figure 19).
\latex{ 1 }
\latex{ -1 }
\latex{ y }
\latex{ x }
Figure 19
Example 5
Let us plot the graph of the function
\latex{g:\left(\R\setminus \left\{k\times \frac{\pi }{2}\mid k\in \Z \right\} \right)\rightarrow \R,g\left(x\right) =\cot x-\tan x}.
Solution
Let us use the definitions and let us perform identical transformations.
\latex{g\left(x\right) =\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}=\frac{\cos ^2x-\sin ^2x}{\sin x\times \cos x}=\frac{2\times\cos 2x}{\sin 2x}=2\times \cot 2x }.
So the graph of \latex{ g } can be derived from the graph of the cotangent function by shrinking it perpendicularly to the \latex{ y }-axis by half, and then by stretching the graph obtained perpendicularly to the \latex{ x }-axis by a factor of \latex{ 2 } (Figure 20).
\latex{\frac{\pi}{2} }
\latex{\pi }
\latex{\frac{\pi}{4} }
\latex{-\frac{\pi}{4} }
\latex{-\frac{\pi}{2} }
\latex{-\pi }
\latex{-\frac{3\pi}{4} }
\latex{\frac{3\pi}{4} }
\latex{ 1 }
\latex{ -1 }
\latex{ y }
\latex{ y }
\latex{ x }
Figure 20
Exercises
{{exercise_number}}. Define the function arccos as the inverse of the cosine function restricted to the interval \latex{\left[0;\pi \right] }, and plot its graph.
- \latex{f:\R\rightarrow \R,f(x)=\cos ^2x-\sin ^2x}
- \latex{g:\R\rightarrow \R,g(x)=\sin x-\cos x}
- \latex{f:\R\rightarrow \R,f(x)=\sin x+\sqrt{3}\times \cos x }
- \latex{h:\R\rightarrow \R,h(x)=\frac{1}{2} \times \sin x+\frac{\sqrt{3} }{2} \times \cos x}
- \latex{g:\R\rightarrow \R,g(x)=\sin ^4x+\cos ^4x}
- \latex{h:\R\rightarrow \R,h(x)=\sin x\times \left|\cos x\right| }
{{exercise_number}}. Characterise the following functions and plot their graphs:
- \latex{f:\left]-\frac{\pi }{2};\frac{\pi }{2} \right[\rightarrow \R,f(x) =\frac{1-\cos ^2x+\sin ^2x}{1+\cos ^2x+\sin ^2x} };
- \latex{g:\left]-3\times \frac{\pi }{4};\frac{\pi }{4} \right[\rightarrow \R,g(x) =\frac{1+\sin 2x}{\cos 2x} }.