Mister Exam
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How to use it?
- Equation:
-
Equation 2*x^2+6*x+9=0
-
Equation 2^2-x-1=x^2-5*x-(-1-x^2)
-
Equation 10(x-9)=7
-
Equation (x+2)^4+(x+2)^2-12=0
- Express {x} in terms of y where:
- 4*x-18*y=-14
- -14*x-1*y=3
- -6*x+2*y=-10
- 10*x-18*y=6
- Sum of series:
-
1/2
- Integral of d{x}:
-
1/2
- Derivative of:
-
1/2
-
Identical expressions
- sin(x/ three +пи/ six)= one / two
- sinus of (x divide by 3 plus пи divide by 6) equally 1 divide by 2
- sinus of (x divide by three plus пи divide by six) equally one divide by two
- sinx/3+пи/6=1/2
- sin(x divide by 3+пи divide by 6)=1 divide by 2
-
Similar expressions
- sin(x/3-пи/6)=1/2
sin(x/3+пи/6)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ sin|- + --| = 1/2 \3 6 /
$$\sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \frac{1}{2}$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n$$
$$\frac{x}{3} = 2 \pi n + \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n$$
$$x_{2} = 6 \pi n + 2 \pi$$
$$\sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n$$
$$\frac{x}{3} = 2 \pi n + \frac{2 \pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n$$
$$x_{2} = 6 \pi n + 2 \pi$$
Sum and product of roots
[src]
sum
2*pi
$$2 \pi$$
=
2*pi
$$2 \pi$$
product
0*2*pi
$$0 \cdot 2 \pi$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 0.0
x2 = -75.398223686155
x3 = -50.2654824574367
x4 = 81.6814089933346
x5 = -37.6991118430775
x6 = 25.1327412287183
x7 = -14771.7686571792
x8 = -18.8495559215388
x9 = -56.5486677646163
x10 = 18.8495559215388
x11 = 56.5486677646163
x12 = -219.911485751286
x13 = 6.28318530717959
x14 = 37.6991118430775
x15 = 100.530964914873
x16 = 94.2477796076938
x17 = -12.5663706143592
x18 = -94.2477796076938
x19 = -131.946891450771
x20 = -163.362817986669
x21 = 1118.40698467797
x22 = 43.9822971502571
x23 = 75.398223686155
x24 = 62.8318530717959
x25 = -87.9645943005142
x26 = -69.1150383789755
x27 = -31.4159265358979
x27 = -31.4159265358979