Mister Exam
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- sin(x/ four -pi/ eight)= zero
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- sinx/4-pi/8=0
- sin(x/4-pi/8)=O
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Similar expressions
- sin(x/4+pi/8)=0
sin(x/4-pi/8)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ sin|- - --| = 0 \4 8 /
$$\sin{\left(\frac{x}{4} - \frac{\pi}{8} \right)} = 0$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{4} - \frac{\pi}{8} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(\frac{x}{4} - \frac{\pi}{8} \right)} = 0$$
The equation is transformed to
$$\cos{\left(\frac{x}{4} + \frac{3 \pi}{8} \right)} = 0$$
This equation is transformed to
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{3 \pi}{8}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = \pi n + \frac{\pi}{8}$$
$$\frac{x}{4} = \pi n - \frac{7 \pi}{8}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{2}$$
$$x_{2} = 4 \pi n - \frac{7 \pi}{2}$$
$$\sin{\left(\frac{x}{4} - \frac{\pi}{8} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(\frac{x}{4} - \frac{\pi}{8} \right)} = 0$$
Divide both parts of the equation by -1
The equation is transformed to
$$\cos{\left(\frac{x}{4} + \frac{3 \pi}{8} \right)} = 0$$
This equation is transformed to
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n + \frac{\pi}{2}$$
$$\frac{x}{4} + \frac{3 \pi}{8} = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{3 \pi}{8}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = \pi n + \frac{\pi}{8}$$
$$\frac{x}{4} = \pi n - \frac{7 \pi}{8}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{2}$$
$$x_{2} = 4 \pi n - \frac{7 \pi}{2}$$
Rapid solution
[src]
pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
9*pi
x2 = ----
2
$$x_{2} = \frac{9 \pi}{2}$$
x2 = 9*pi/2
Sum and product of roots
[src]
sum
pi 9*pi -- + ---- 2 2
$$\frac{\pi}{2} + \frac{9 \pi}{2}$$
=
5*pi
$$5 \pi$$
product
pi 9*pi --*---- 2 2
$$\frac{\pi}{2} \frac{9 \pi}{2}$$
=
2 9*pi ----- 4
$$\frac{9 \pi^{2}}{4}$$
9*pi^2/4
Numerical answer
[src]
x1 = 39.2699081698724
x2 = 103586.163770489
x3 = -86.3937979737193
x4 = -98.9601685880785
x5 = -174.358392274234
x6 = 14.1371669411541
x7 = -48.6946861306418
x8 = 139.800873084746
x9 = -589.048622548086
x10 = 1.5707963267949
x11 = -450.818545790135
x12 = 51.8362787842316
x13 = -10.9955742875643
x14 = 26.7035375555132
x15 = -23.5619449019235
x16 = 190.066355542182
x17 = 102.101761241668
x18 = 76.9690200129499
x19 = -36.1283155162826
x20 = -73.8274273593601
x21 = 64.4026493985908
x22 = 89.5353906273091
x23 = -61.261056745001
x23 = -61.261056745001