Mister Exam
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Identical expressions
- cos(x/ two +pi/ four)= zero
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Similar expressions
- cos(x/2-pi/4)=0
cos(x/2+pi/4)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ cos|- + --| = 0 \2 4 /
$$\cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = 0$$
Detail solution
Given the equation
$$\cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{4} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{\pi}{4}$$
$$\frac{x}{2} = \pi n - \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{3 \pi}{2}$$
$$\cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{4} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{\pi}{4}$$
$$\frac{x}{2} = \pi n - \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{3 \pi}{2}$$
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 2 2
$$\frac{\pi}{2} + \frac{5 \pi}{2}$$
=
3*pi
$$3 \pi$$
product
pi 5*pi --*---- 2 2
$$\frac{\pi}{2} \frac{5 \pi}{2}$$
=
2 5*pi ----- 4
$$\frac{5 \pi^{2}}{4}$$
5*pi^2/4
Rapid solution
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
5*pi
x2 = ----
2
$$x_{2} = \frac{5 \pi}{2}$$
x2 = 5*pi/2
Numerical answer
[src]
x1 = -29.845130209103
x2 = -67.5442420521806
x3 = 64.4026493985908
x4 = -36.1283155162826
x5 = -92.6769832808989
x6 = -61.261056745001
x7 = -98.9601685880785
x8 = 32.9867228626928
x9 = -4.71238898038469
x10 = -10.9955742875643
x11 = 26.7035375555132
x12 = -105.243353895258
x13 = 95.8185759344887
x14 = -17.2787595947439
x15 = -86.3937979737193
x16 = -48.6946861306418
x17 = 76.9690200129499
x18 = 45.553093477052
x19 = -2913.82718620453
x20 = -80.1106126665397
x21 = 7.85398163397448
x22 = 102.101761241668
x23 = 14.1371669411541
x24 = -563.915881319368
x25 = 70.6858347057703
x26 = 83.2522053201295
x27 = 51.8362787842316
x28 = 20.4203522483337
x29 = 1.5707963267949
x30 = 89.5353906273091
x31 = 58.1194640914112
x32 = -218.340689424491
x33 = -54.9778714378214
x34 = -42.4115008234622
x35 = -256.039801267568
x36 = 39.2699081698724
x37 = -23.5619449019235
x38 = -73.8274273593601
x38 = -73.8274273593601