Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2-8*x+17=0
-
Equation 2-3(7+2x)=11
-
Equation x^2-2x+1=0
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Equation sin(x/2)=1/2
- Express {x} in terms of y where:
- 7*x+19*y=12
- 17*x+20*y=-12
- 18*x-11*y=2
- 2*x-8*y=4
- Derivative of:
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sin(x/2)
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1/2
- Graphing y =:
- sin(x/2)
- Integral of d{x}:
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sin(x/2)
-
1/2
- Sum of series:
-
1/2
-
Identical expressions
- sin(x/ two)= one / two
- sinus of (x divide by 2) equally 1 divide by 2
- sinus of (x divide by two) equally one divide by two
- sinx/2=1/2
- sin(x divide by 2)=1 divide by 2
-
Similar expressions
- 2sinx/2=√2
- b+1/(2*(cos(x)+1))-(cos(x/2)^(2)*log(cos(x/2)))/(cos(x)+1)+(cos(x/2)^(2)*log(sin(x/2)))/(cos(x)+1)=y
- 2sin*x/2=√3
sin(x/2)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(\frac{x}{2} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{5 \pi}{3}$$
$$\sin{\left(\frac{x}{2} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{5 \pi}{3}$$
Rapid solution
[src]
pi
x1 = --
3
$$x_{1} = \frac{\pi}{3}$$
5*pi
x2 = ----
3
$$x_{2} = \frac{5 \pi}{3}$$
x2 = 5*pi/3
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 3 3
$$\frac{\pi}{3} + \frac{5 \pi}{3}$$
=
2*pi
$$2 \pi$$
product
pi 5*pi --*---- 3 3
$$\frac{\pi}{3} \frac{5 \pi}{3}$$
=
2 5*pi ----- 9
$$\frac{5 \pi^{2}}{9}$$
5*pi^2/9
Numerical answer
[src]
x1 = 13.6135681655558
x2 = 5.23598775598299
x3 = -7.33038285837618
x4 = -74.3510261349584
x5 = -82.7286065445312
x6 = 101.57816246607
x7 = -70.162235930172
x8 = 93.2005820564972
x9 = -19.8967534727354
x10 = 42.9350995990605
x11 = -32.4631240870945
x12 = 55.5014702134197
x13 = -45.0294947014537
x14 = 30.3687289847013
x15 = -11.5191730631626
x16 = -61.7846555205993
x17 = 59100.6881968824
x18 = -99.4837673636768
x19 = -36.6519142918809
x20 = -24.0855436775217
x21 = 38.7463093942741
x22 = 1.0471975511966
x23 = 331.961623729322
x24 = -57.5958653158129
x25 = 68.0678408277789
x26 = -49.2182849062401
x27 = 26.1799387799149
x28 = 63.8790506229925
x29 = 114.144533080429
x30 = -86.9173967493176
x31 = -346.622389446074
x32 = -95.2949771588904
x33 = 17.8023583703422
x34 = 51.3126800086333
x35 = 76.4454212373516
x36 = 105.766952670856
x37 = 80.634211442138
x38 = 89.0117918517108
x38 = 89.0117918517108
The graph