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Trigonometric equations, inequalities
(extra-curricular topic)
The properties of the functions can often be use well while solving trigonometric equations or inequalities. The following few exercises give examples for this.
Example 1
Let us solve the following equation on the set of real numbers:
\latex{\sin{x}+2\times\sin{2x}+\cos{4x}=4}.
Solution
An \latex{\R\rightarrow\R} type function can be defined with the expression on the left hand-side of the equation:
\latex{f:\R\rightarrow\R,\;\;f(x)=\sin{x}+2\times\sin 2x+\cos{4x}}.
\latex{ 4 } is an upper bound of the function \latex{ f } , since the first and the last term of the right-hand-side are at most 1 and its second term is at most \latex{ 2 }. The question is whether \latex{ f } takes the value of \latex{ 4 } somewhere, and if so where.
Let us assume that there is an \latex{x\in\R} for which \latex{f(x) = 4}.
In this case for the first term \latex{\sin x = 1}, so
\latex{\begin{equation}x=\frac{\pi}{2}+2k\pi\;\;(k\in\Z).\;\;\;\;\;\;\;\;\;\;\;\;\end{equation}}
In the second term \latex{\sin 2x = 1}, so \latex{2x = \frac{\pi}{2}+ 2n\pi},
\latex{\begin{equation}x=\frac{\pi}{4}+n\pi\;\;(n\in\Z).\;\;\;\;\;\;\;\;\;\;\;\;\end{equation}}
Finally \latex{\cos 4x = 1}, then \latex{4x = 2l\pi},
\latex{\begin{equation}x=l\times\frac{\pi}{2}\;\;(l\in\Z).\;\;\;\;\;\;\;\;\;\;\;\;\end{equation}}
According to the assumption both (1) and (2) are true, so there are integers \latex{ k } and \latex{ n } for which
\latex{\frac{\pi}{2}+2k\pi=\frac{\pi}4+n\pi},
which implies the following when multiplied by \latex{ 4 } and divided by \latex{\pi}:
\latex{2+8k=1+4n.}
There is an even number on the left-hand-side, the right-hand-side is odd, and so this is a contradiction. Thus there is no such \latex{ x } for which the value of all three terms is maximized, i.e. the equation has no roots.
Example 2
How many solutions does the following equation have on the set of real numbers:
\latex{\lvert\sin x\rvert=\frac{2}{2,015\times\pi}\times x. }
Solution
It is again worth defining two functions:
\latex{f:\R\rightarrow\R,\;\;f(x)=\lvert\sin x\rvert} and
\latex{g:\R\rightarrow\R,\;\;g(x)=\frac{2}{2,015\times\pi}\times x}.
\latex{g:\R\rightarrow\R,\;\;g(x)=\frac{2}{2,015\times\pi}\times x}.
The graphs of the functions \latex{ f } and \latex{ g } in one coordinate system: (Figure 21)
\latex{f(x)=\left|\sin x\right|}
\latex{g(x)=\frac{2}{2,015\times\pi}\times x}
\latex{\pi}
\latex{-\pi}
\latex{-2\pi}
\latex{-3\pi}
\latex{2\pi}
\latex{1,007\pi}
\latex{\frac{2015\times\pi}{2}}
\latex{1,008\pi}
\latex{ 1 }
\latex{ y }
\latex{ x }
Figure 21
The values of \latex{ f } are non-negative real numbers; the values of \latex{ g } are negative for negative values of \latex{ x }, so the equation does not have a negative root.
Since\latex{f(x) = 0}, if \latex{x = k\pi\;\; (k\in\Z)} and \latex{g(0) = 0}, thus \latex{x = 0} is a root of the equation. We accept (it can be proven) that for \latex{0 \leq x \leq \pi} the graphs of \latex{ f } and \latex{ g } intersect each other at one more place, since \latex{f(\pi) = 0} and \latex{g(\pi) \gt 0}, while\latex{f\left(\frac{\pi}{2}\right) = 1} and \latex{g\left(\frac{\pi}{2}\right) \lt 1}.
Since\latex{f(x) = 0}, if \latex{x = k\pi\;\; (k\in\Z)} and \latex{g(0) = 0}, thus \latex{x = 0} is a root of the equation. We accept (it can be proven) that for \latex{0 \leq x \leq \pi} the graphs of \latex{ f } and \latex{ g } intersect each other at one more place, since \latex{f(\pi) = 0} and \latex{g(\pi) \gt 0}, while\latex{f\left(\frac{\pi}{2}\right) = 1} and \latex{g\left(\frac{\pi}{2}\right) \lt 1}.
It can similarly be thought over that while the value of \latex{ g } does not reach \latex{ 1 }, on every interval of the form \latex{[k\pi; (k + 1)\pi]} the values of \latex{ f } and \latex{ g } will be equal at \latex{ 2 } places, so the equation has two roots.
\latex{g(x) =\frac{2}{2,015\times\pi}\times x = 1} is satisfied if \latex{x =\frac{2}{2,015\times\pi}} , which is also the midpoint of the interval \latex{[1,007 \times\pi; 1,008 \times \pi]}. The values of \latex{ f } and \latex{ g } are equal at two places on this interval too, after this for larger values of \latex{ x } \latex{\;g(x) \gt f(x)}. So the equation has a total of \latex{ 2,016 } roots.
\latex{g(x) =\frac{2}{2,015\times\pi}\times x = 1} is satisfied if \latex{x =\frac{2}{2,015\times\pi}} , which is also the midpoint of the interval \latex{[1,007 \times\pi; 1,008 \times \pi]}. The values of \latex{ f } and \latex{ g } are equal at two places on this interval too, after this for larger values of \latex{ x } \latex{\;g(x) \gt f(x)}. So the equation has a total of \latex{ 2,016 } roots.
Example 3
Let us solve the equation \latex{x+\frac{1}{x}=2\times\sin{\left(\frac{\pi}{2}\times x\right)}} on the set of real numbers.
Solution
We define functions on the set of real numbers not equal to \latex{ 0 } with the expressions on the two sides of the equation:
\latex{f:(\R\backslash\left\{0\right\})\rightarrow\R, \;\;f(x)=x+\frac{1}{x}},
and
\latex{g:(\R\backslash\left\{0\right\})\rightarrow\R,\;\;g(x)=2\times\left(\frac{\pi}{2}\times x\right)}.
The function f is familiar from year \latex{ 9 }. On the set of positive numbers its minimum is \latex{ 2 }, on the set of negative numbers its maximum is \latex{ –2 }.
The maximum of the function \latex{ g } is \latex{ 2 }, its minimum is \latex{ –2 }. This way their values can be equal only in \latex{ 2 } cases at most:
On the set of positive numbers\latex{f(x) \geq2}, and it is \latex{ 2 } only if \latex{x = 1}. At this place\latex{g(1) = 2 \times \sin = 2}, so\latex{x = 1} is a root of the equation.
On the set of negative numbers \latex{f(x) \leq –2}, and it is \latex{ –2 } only if \latex{x = -1}.
At this place \latex{g(–1) = 2 \times \sin = –2}, so \latex{x = -1} is also a root of the equation.
There are no other roots. The sketch image of the graphs of the two functions can be seen in Figure 22.
The maximum of the function \latex{ g } is \latex{ 2 }, its minimum is \latex{ –2 }. This way their values can be equal only in \latex{ 2 } cases at most:
On the set of positive numbers\latex{f(x) \geq2}, and it is \latex{ 2 } only if \latex{x = 1}. At this place\latex{g(1) = 2 \times \sin = 2}, so\latex{x = 1} is a root of the equation.
On the set of negative numbers \latex{f(x) \leq –2}, and it is \latex{ –2 } only if \latex{x = -1}.
At this place \latex{g(–1) = 2 \times \sin = –2}, so \latex{x = -1} is also a root of the equation.
There are no other roots. The sketch image of the graphs of the two functions can be seen in Figure 22.
\latex{f(x)=x+\frac 1 x}
\latex{g(x)=2\times\sin{\left(\frac{\pi}2\times x\right)}}
\latex{ 2 }
\latex{ 1 }
\latex{ -1 }
\latex{ -2 }
y
x
Figure 22
Example 4
Let us represent the set of the points with coordinates \latex{(x; y)} in the Cartesian coordinate system for which:
- \latex{\sin x=\sin y};
- \latex{\sin^2x+\cos^2y=2.}
Solution (a)
The definition and the properties of the sine function imply that equality holds if and only if
\latex{y=x+2k\pi\;\;(k\in\Z)} or \latex{y=\pi-x+2n\pi\;\;(n\in\Z),\;\;\;\;\;\;\;\;\;\;\;\;\tag 1}
i.e.
\latex{y=-x+(2n+1)\pi.\;\;\;\;\;\;\;\;\;\;\;\;\tag 2}
The equations (1) describe a set of straight lines parallel with the angle bisector at \latex{ 45º }, and the equations (2) describe a set of straight lines perpendicular to these. (Figure 23)
Solution (b)
As \latex{|\sin x|\leq 1} and \latex{|\cos y|\leq 1}, equality holds only if
\latex{\lvert\sin x\rvert=1,} i.e. \latex{x=\frac{\pi}{2}+k\pi\;\;(k\in\Z)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\tag 1}
and
\latex{\lvert\cos x\rvert=1,} i.e. \latex{y=n\pi\;\;(n\in\Z)\;\;\;\;\;\;\;\;\;\;\;\;\tag 2}
The equalities (1) describe the points of a set of straight lines parallel with the \latex{ y }-axis, and the equalities (2) describe the points of the \latex{ x }-axis with the points of a set of straight lines parallel with the \latex{ x }-axis inclusive. Both equalities hold for the intersection points of the straight lines and only there. (Figure 24)
\latex{y=x}
\latex{y=x+2\pi}
\latex{y=-x-2\pi}
\latex{y=-x-3\pi}
\latex{y=-x-\pi}
\latex{y=-x+\pi}
\latex{y=-x+3\pi}
\latex{3\pi}
\latex{2\pi}
\latex{\pi}
\latex{\pi}
\latex{2\pi}
\latex{3\pi}
\latex{-3\pi}
\latex{-2\pi}
\latex{-\pi}
\latex{-\pi}
\latex{-2\pi}
\latex{-3\pi}
\latex{ y }
Figure 23
\latex{2\pi}
\latex{\pi}
\latex{\frac{\pi}2}
\latex{3\pi}
\latex{\frac{-3\pi}2}
\latex{-\pi}
\latex{-2\pi}
\latex{-3\pi}
\latex{\frac{-5\pi}2}
\latex{-\frac{\pi}2}
\latex{\frac{3\pi}2}
\latex{\frac{5\pi}2}
\latex{ y }
Figure 24
Example 5
Let us solve the following inequality on the set of real numbers:
\latex{\tan^2{x}+\cot^2x\leq\sqrt2\times(\sin x+\cos x)}.
Solution
According to example \latex{ 2 } of the \latex{ 3 }rd point of this chapter the expression on the right-hand-side can be rewritten as follows:
\latex{\sqrt2\times(\sin x+\cos x)=2\times\sin{\left(x+\frac{\pi}{4}\right)}}.
Following the way we got used to we can define two functions with the expressions on the right-hand-side and on the left-hand-side; we are not going to write down their definitions in detail. The value of the right-hand-side is \latex{ 2 } at most, and it takes it if
\latex{x+\frac{\pi}{4}=\frac{\pi}{2}+2k\pi\;\;(k\in\Z),} i.e.
\latex{x=\frac{\pi}{4}+2k\pi\;\;(k\in\Z)}.
\latex{x=\frac{\pi}{4}+2k\pi\;\;(k\in\Z)}.
The expression on the left-hand-side can be rewritten as follows:
\latex{\tan^2x+\frac{1}{\tan^2 x}}.
As \latex{\tan^2x\neq0}, \latex{\tan^2x \gt 0}, so its minimum is \latex{ 2 }, and it takes it only if \latex{\tan^2x = 1}. It is satisfied for the values of \latex{ x } obtained previously.
So we get that all the solutions of the inequality are the real numbers in the form of
\latex{x=\frac{\pi}{4}+2k\pi\;\;(k\in\Z).}
Exercises
{{exercise_number}}. How many solutions does the equation \latex{\log_{\frac{5}{2}\times \pi}x=\cos x} have on the set of positive real numbers?
{{exercise_number}}. For which values of the real parameter a does the equation \latex{\sin^4x + \cos^4x = a} have real roots?
{{exercise_number}}. Give the range of the function \latex{f: (\R\backslash \left\{k |k\in\Z\right\}) \rightarrow\R, f(x) = \tan x + \cot x}.
{{exercise_number}}. Solve the following equation on the set of real numbers:
\latex{\left(\cos 2x-\cos 4x\right)^2=4+\cos^2{3x}}.
{{exercise_number}}. Represent the set of the points in the Cartesian coordinate system for which:
- \latex{\tan x=\tan y};
- \latex{\sin^2 x+\cos^2 y=0}.