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Plotting a function graph/ sin(x)+cos(x)

Graphing y = sin(x)+cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(x) + cos(x)
f(x)=sin⁡(x)+cos⁡(x)f{\left(x \right)} = \sin{\left(x \right)} + \cos{\left(x \right)} 
f = sin(x) + cos(x)
The graph of the function
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.05-5
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:
sin⁡(x)+cos⁡(x)=0\sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
Numerical solution
x1=−0.785398163397448x_{1} = -0.785398163397448
x2=−51.0508806208341x_{2} = -51.0508806208341
x3=−98.174770424681x_{3} = -98.174770424681
x4=−7.06858347057703x_{4} = -7.06858347057703
x5=−85.6083998103219x_{5} = -85.6083998103219
x6=84.037603483527x_{6} = 84.037603483527
x7=−95.0331777710912x_{7} = -95.0331777710912
x8=18.0641577581413x_{8} = 18.0641577581413
x9=77.7544181763474x_{9} = 77.7544181763474
x10=−107.59954838545x_{10} = -107.59954838545
x11=36.9137136796801x_{11} = 36.9137136796801
x12=−41.6261026600648x_{12} = -41.6261026600648
x13=−32.2013246992954x_{13} = -32.2013246992954
x14=46.3384916404494x_{14} = 46.3384916404494
x15=8.63937979737193x_{15} = 8.63937979737193
x16=−76.1836218495525x_{16} = -76.1836218495525
x17=−54.1924732744239x_{17} = -54.1924732744239
x18=40.0553063332699x_{18} = 40.0553063332699
x19=−25.9181393921158x_{19} = -25.9181393921158
x20=55.7632696012188x_{20} = 55.7632696012188
x21=71.4712328691678x_{21} = 71.4712328691678
x22=−60.4756585816035x_{22} = -60.4756585816035
x23=21.2057504117311x_{23} = 21.2057504117311
x24=−38.484510006475x_{24} = -38.484510006475
x25=2.35619449019234x_{25} = 2.35619449019234
x26=99.7455667514759x_{26} = 99.7455667514759
x27=351.072979038659x_{27} = 351.072979038659
x28=30.6305283725005x_{28} = 30.6305283725005
x29=90.3207887907066x_{29} = 90.3207887907066
x30=11.7809724509617x_{30} = 11.7809724509617
x31=−57.3340659280137x_{31} = -57.3340659280137
x32=−29.0597320457056x_{32} = -29.0597320457056
x33=−88.7499924639117x_{33} = -88.7499924639117
x34=−79.3252145031423x_{34} = -79.3252145031423
x35=65.1880475619882x_{35} = 65.1880475619882
x36=27.4889357189107x_{36} = 27.4889357189107
x37=−44.7676953136546x_{37} = -44.7676953136546
x38=−16.4933614313464x_{38} = -16.4933614313464
x39=−22.776546738526x_{39} = -22.776546738526
x40=−19.6349540849362x_{40} = -19.6349540849362
x41=−91.8915851175014x_{41} = -91.8915851175014
x42=−63.6172512351933x_{42} = -63.6172512351933
x43=−3.92699081698724x_{43} = -3.92699081698724
x44=49.4800842940392x_{44} = 49.4800842940392
x45=−13.3517687777566x_{45} = -13.3517687777566
x46=−69.9004365423729x_{46} = -69.9004365423729
x47=93.4623814442964x_{47} = 93.4623814442964
x48=−73.0420291959627x_{48} = -73.0420291959627
x49=87.1791961371168x_{49} = 87.1791961371168
x50=52.621676947629x_{50} = 52.621676947629
x51=−66.7588438887831x_{51} = -66.7588438887831
x52=24.3473430653209x_{52} = 24.3473430653209
x53=33.7721210260903x_{53} = 33.7721210260903
x54=−10.2101761241668x_{54} = -10.2101761241668
x55=−35.3429173528852x_{55} = -35.3429173528852
x56=5.49778714378214x_{56} = 5.49778714378214
x57=58.9048622548086x_{57} = 58.9048622548086
x58=96.6039740978861x_{58} = 96.6039740978861
x59=43.1968989868597x_{59} = 43.1968989868597
x60=14.9225651045515x_{60} = 14.9225651045515
x61=74.6128255227576x_{61} = 74.6128255227576
x62=−82.4668071567321x_{62} = -82.4668071567321
x63=62.0464549083984x_{63} = 62.0464549083984
x64=−47.9092879672443x_{64} = -47.9092879672443
x65=68.329640215578x_{65} = 68.329640215578
x66=80.8960108299372x_{66} = 80.8960108299372
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) + cos(x).
sin⁡(0)+cos⁡(0)\sin{\left(0 \right)} + \cos{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−sin⁡(x)+cos⁡(x)=0- \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
The values of the extrema at the points:
 pi    ___ 
(--, \/ 2 )
 4


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]\left(-\infty, \frac{\pi}{4}\right]
Increasing at intervals
[π4,∞)\left[\frac{\pi}{4}, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−(sin⁡(x)+cos⁡(x))=0- (\sin{\left(x \right)} + \cos{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,−π4]\left(-\infty, - \frac{\pi}{4}\right]
Convex at the intervals
[−π4,∞)\left[- \frac{\pi}{4}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x) + cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)+cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
sin⁡(x)+cos⁡(x)=−sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} = - \sin{\left(x \right)} + \cos{\left(x \right)}
- No
sin⁡(x)+cos⁡(x)=sin⁡(x)−cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} = \sin{\left(x \right)} - \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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