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  • How to use it?

  • Graphing y =:
  • x*e^-x
  • x^3-(x^4/4)
  • |x+4|
  • -x^4+2*x^2+8
  • Identical expressions

  • sin(five *x)^ two /sin(x)^ two -cos(five *x)^ two /cos(x)^ two
  • sinus of (5 multiply by x) squared divide by sinus of (x) squared minus co sinus of e of (5 multiply by x) squared divide by co sinus of e of (x) squared
  • sinus of (five multiply by x) to the power of two divide by sinus of (x) to the power of two minus co sinus of e of (five multiply by x) to the power of two divide by co sinus of e of (x) to the power of two
  • sin(5*x)2/sin(x)2-cos(5*x)2/cos(x)2
  • sin5*x2/sinx2-cos5*x2/cosx2
  • sin(5*x)²/sin(x)²-cos(5*x)²/cos(x)²
  • sin(5*x) to the power of 2/sin(x) to the power of 2-cos(5*x) to the power of 2/cos(x) to the power of 2
  • sin(5x)^2/sin(x)^2-cos(5x)^2/cos(x)^2
  • sin(5x)2/sin(x)2-cos(5x)2/cos(x)2
  • sin5x2/sinx2-cos5x2/cosx2
  • sin5x^2/sinx^2-cos5x^2/cosx^2
  • sin(5*x)^2 divide by sin(x)^2-cos(5*x)^2 divide by cos(x)^2
  • Similar expressions

  • sin(5*x)^2/sin(x)^2+cos(5*x)^2/cos(x)^2
  • sin(5*x)^2/sinx^2-cos(5*x)^2/cosx^2

Graphing y = sin(5*x)^2/sin(x)^2-cos(5*x)^2/cos(x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
          2           2     
       sin (5*x)   cos (5*x)
f(x) = --------- - ---------
           2           2    
        sin (x)     cos (x) 
$$f{\left(x \right)} = - \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}$$
f = -cos(5*x)^2/cos(x)^2 + sin(5*x)^2/sin(x)^2
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = 4.71238898038469$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = - \frac{5 \pi}{6}$$
$$x_{2} = - \frac{3 \pi}{4}$$
$$x_{3} = - \frac{2 \pi}{3}$$
$$x_{4} = - \frac{\pi}{3}$$
$$x_{5} = - \frac{\pi}{4}$$
$$x_{6} = - \frac{\pi}{6}$$
$$x_{7} = \frac{\pi}{6}$$
$$x_{8} = \frac{\pi}{4}$$
$$x_{9} = \frac{\pi}{3}$$
$$x_{10} = \frac{2 \pi}{3}$$
$$x_{11} = \frac{3 \pi}{4}$$
$$x_{12} = \frac{5 \pi}{6}$$
Numerical solution
$$x_{1} = 82.2050077689329$$
$$x_{2} = 85.870199198121$$
$$x_{3} = 74.3510261349584$$
$$x_{4} = 32.2013246992954$$
$$x_{5} = 10.2101761241668$$
$$x_{6} = -32.2013246992954$$
$$x_{7} = -41.8879020478639$$
$$x_{8} = 44.5058959258554$$
$$x_{9} = -68.0678408277789$$
$$x_{10} = -90.0589894029074$$
$$x_{11} = 12.0427718387609$$
$$x_{12} = -38.2227106186758$$
$$x_{13} = -49.7418836818384$$
$$x_{14} = -75.9218224617533$$
$$x_{15} = 52.3598775598299$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = 82.4668071567321$$
$$x_{18} = -103.148958792865$$
$$x_{19} = -69.9004365423729$$
$$x_{20} = -46.0766922526503$$
$$x_{21} = -2.0943951023932$$
$$x_{22} = -77.7544181763474$$
$$x_{23} = 16.4933614313464$$
$$x_{24} = 71.733032256967$$
$$x_{25} = 70.162235930172$$
$$x_{26} = 4.18879020478639$$
$$x_{27} = -91.8915851175014$$
$$x_{28} = 93.7241808320955$$
$$x_{29} = -74.6128255227576$$
$$x_{30} = 38.2227106186758$$
$$x_{31} = 41.8879020478639$$
$$x_{32} = -55.7632696012188$$
$$x_{33} = 88.4881930761125$$
$$x_{34} = 34.0339204138894$$
$$x_{35} = -13.6135681655558$$
$$x_{36} = -65.1880475619882$$
$$x_{37} = -31.9395253114962$$
$$x_{38} = 60.2138591938044$$
$$x_{39} = -85.870199198121$$
$$x_{40} = 90.3207887907066$$
$$x_{41} = 48.1710873550435$$
$$x_{42} = -17.8023583703422$$
$$x_{43} = -35.6047167406843$$
$$x_{44} = -16.2315620435473$$
$$x_{45} = -93.7241808320955$$
$$x_{46} = 98.174770424681$$
$$x_{47} = 5.75958653158129$$
$$x_{48} = 76.1836218495525$$
$$x_{49} = -83.7758040957278$$
$$x_{50} = 63.8790506229925$$
$$x_{51} = -79.5870138909414$$
$$x_{52} = 96.342174710087$$
$$x_{53} = -82.2050077689329$$
$$x_{54} = -21.4675497995303$$
$$x_{55} = -25.6563400043166$$
$$x_{56} = -60.2138591938044$$
$$x_{57} = 40.0553063332699$$
$$x_{58} = -33.7721210260903$$
$$x_{59} = 22.5147473507269$$
$$x_{60} = 8.37758040957278$$
$$x_{61} = -25.9181393921158$$
$$x_{62} = 27.7507351067098$$
$$x_{63} = -61.7846555205993$$
$$x_{64} = 56.025068989018$$
$$x_{65} = -87.1791961371168$$
$$x_{66} = -24.0855436775217$$
$$x_{67} = 78.0162175641465$$
$$x_{68} = 1.0471975511966$$
$$x_{69} = 46.6002910282486$$
$$x_{70} = -39.7935069454707$$
$$x_{71} = -9.94837673636768$$
$$x_{72} = -99.7455667514759$$
$$x_{73} = -51.3126800086333$$
$$x_{74} = 62.0464549083984$$
$$x_{75} = 54.1924732744239$$
$$x_{76} = 68.329640215578$$
$$x_{77} = 49.7418836818384$$
$$x_{78} = 24.0855436775217$$
$$x_{79} = 18.0641577581413$$
$$x_{80} = -97.9129710368819$$
$$x_{81} = 100.007366139275$$
$$x_{82} = -53.9306738866248$$
$$x_{83} = -43.1968989868597$$
$$x_{84} = 30.3687289847013$$
$$x_{85} = 92.1533845053006$$
$$x_{86} = -11.7809724509617$$
$$x_{87} = 84.2994028713261$$
$$x_{88} = 66.497044500984$$
$$x_{89} = -57.5958653158129$$
$$x_{90} = -63.8790506229925$$
$$x_{91} = 26.1799387799149$$
$$x_{92} = 16.2315620435473$$
$$x_{93} = -71.733032256967$$
$$x_{94} = -5.75958653158129$$
$$x_{95} = 19.8967534727354$$
$$x_{96} = 14.9225651045515$$
$$x_{97} = -47.9092879672443$$
$$x_{98} = -27.7507351067098$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(5*x)^2/sin(x)^2 - cos(5*x)^2/cos(x)^2.
$$- \frac{\cos^{2}{\left(0 \cdot 5 \right)}}{\cos^{2}{\left(0 \right)}} + \frac{\sin^{2}{\left(0 \cdot 5 \right)}}{\sin^{2}{\left(0 \right)}}$$
The result:
$$f{\left(0 \right)} = \text{NaN}$$
- the solutions of the equation d'not exist
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = 4.71238898038469$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(5*x)^2/sin(x)^2 - cos(5*x)^2/cos(x)^2, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}}{x}\right)$$
True

Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}}{x}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}} = - \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}$$
- Yes
$$- \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}} = \frac{\cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(x \right)}}$$
- No
so, the function
is
even