Mister Exam
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Similar expressions
- 2*sin(x/2+pi/6)=1
2*sin(x/2-pi/6)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\
2*sin|- - --| = 1
\2 6 /
$$2 \sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
Detail solution
Given the equation
$$2 \sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{4 \pi}{3}$$
$$2 \sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2
The equation is transformed to
$$\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} + \frac{\pi}{3} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{4 \pi}{3}$$
Rapid solution
[src]
2*pi
x1 = ----
3
$$x_{1} = \frac{2 \pi}{3}$$
x2 = 2*pi
$$x_{2} = 2 \pi$$
x2 = 2*pi
Sum and product of roots
[src]
sum
2*pi ---- + 2*pi 3
$$\frac{2 \pi}{3} + 2 \pi$$
=
8*pi ---- 3
$$\frac{8 \pi}{3}$$
product
2*pi ----*2*pi 3
$$\frac{2 \pi}{3} \cdot 2 \pi$$
=
2 4*pi ----- 3
$$\frac{4 \pi^{2}}{3}$$
4*pi^2/3
Numerical answer
[src]
x1 = -18.8495559215388
x2 = -85.870199198121
x3 = 64.9262481741891
x4 = -56.5486677646163
x5 = 82919.196498849
x6 = -81.6814089933346
x7 = -31.4159265358979
x8 = 94.2477796076938
x9 = 52.3598775598299
x10 = 81.6814089933346
x11 = 77.4926187885482
x12 = 203.15632493214
x13 = 18893.538218689
x14 = 27.2271363311115
x15 = 14.6607657167524
x16 = 6.28318530717959
x17 = -6.28318530717959
x18 = 31.4159265358979
x19 = -23.0383461263252
x20 = 18.8495559215388
x21 = -94.2477796076938
x22 = -596.902604182061
x23 = -60.7374579694027
x24 = 56.5486677646163
x25 = -10.471975511966
x26 = -48.1710873550435
x27 = -98.4365698124802
x28 = -73.3038285837618
x29 = -43.9822971502571
x30 = 2.0943951023932
x31 = 69.1150383789755
x32 = 102.625360017267
x33 = -69.1150383789755
x34 = -123.569311041199
x35 = -35.6047167406843
x36 = 90.0589894029074
x37 = 39.7935069454707
x38 = 43.9822971502571
x38 = 43.9822971502571