Mister Exam
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Identical expressions
- cos((x/ two)-pi/ four)= zero
- co sinus of e of ((x divide by 2) minus Pi divide by 4) equally 0
- co sinus of e of ((x divide by two) minus Pi divide by four) equally zero
- cosx/2-pi/4=0
- cos((x/2)-pi/4)=O
- cos((x divide by 2)-pi divide by 4)=0
-
Similar expressions
- cos((x/2)-(pi/4))=1
- cos((x/2)+pi/4)=0
- cos((x)/(2)-(pi)/(4))=-1
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
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{2} = 2 \pi n + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{\pi}{2}$$
$$x_{2} = 4 \pi n + \frac{3 \pi}{2}$$
$$\cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = 0$$
- 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
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{2} = 2 \pi n + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{\pi}{2}$$
$$x_{2} = 4 \pi n + \frac{3 \pi}{2}$$
Rapid solution
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
3*pi
x2 = ----
2
$$x_{2} = \frac{3 \pi}{2}$$
x2 = 3*pi/2
Sum and product of roots
[src]
sum
pi 3*pi - -- + ---- 2 2
$$- \frac{\pi}{2} + \frac{3 \pi}{2}$$
=
pi
$$\pi$$
product
-pi 3*pi ----*---- 2 2
$$- \frac{\pi}{2} \frac{3 \pi}{2}$$
=
2 -3*pi ------ 4
$$- \frac{3 \pi^{2}}{4}$$
-3*pi^2/4
Numerical answer
[src]
x1 = 17.2787595947439
x2 = -14.1371669411541
x3 = -95.8185759344887
x4 = -146.084058391925
x5 = -26.7035375555132
x6 = 4.71238898038469
x7 = -246.615023306799
x8 = 23.5619449019235
x9 = 42.4115008234622
x10 = 36.1283155162826
x11 = 29.845130209103
x12 = 61.261056745001
x13 = 48.6946861306418
x14 = -7.85398163397448
x15 = -51.8362787842316
x16 = -76.9690200129499
x17 = -89.5353906273091
x18 = -39.2699081698724
x19 = 80.1106126665397
x20 = 67.5442420521806
x21 = -32.9867228626928
x22 = -20.4203522483337
x23 = -824.668071567321
x24 = -45.553093477052
x25 = -1.5707963267949
x26 = -70.6858347057703
x27 = -83.2522053201295
x28 = 98.9601685880785
x29 = -102.101761241668
x30 = 73.8274273593601
x31 = -58.1194640914112
x32 = 92.6769832808989
x33 = 54.9778714378214
x34 = 86.3937979737193
x35 = -64.4026493985908
x36 = 10.9955742875643
x36 = 10.9955742875643