Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -x/12+x=55/12
-
Equation 9/(x-2)=9/2
-
Equation 2-3*(2*x+2)=5-4*x
-
Equation x^4=(4x-21)^2
- Express {x} in terms of y where:
- 4*x+2*y=5
- -10*x-18*y=15
- 2*x-8*y=4
- 2*x-15*y=-1
- Integral of d{x}:
-
sin(x)^2
-
1/2
- Graphing y =:
- sin(x)^2
- Derivative of:
-
sin(x)^2
-
1/2
- Sum of series:
-
1/2
-
Identical expressions
- sin(x)^ two = one / two
- sinus of (x) squared equally 1 divide by 2
- sinus of (x) to the power of two equally one divide by two
- sin(x)2=1/2
- sinx2=1/2
- sin(x)²=1/2
- sin(x) to the power of 2=1/2
- sinx^2=1/2
- sin(x)^2=1 divide by 2
-
Similar expressions
- sinx^2=1/2
sin(x)^2=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin^{2}{\left(x \right)} = \frac{1}{2}$$
transform
$$- \frac{\cos{\left(2 x \right)}}{2} = 0$$
$$\sin^{2}{\left(x \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = - \frac{1}{2}$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{\sqrt{2}}{2}$$
Simplify
$$w_{2} = - \frac{\sqrt{2}}{2}$$
Simplify
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{4}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{4}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{4}$$
$$\sin^{2}{\left(x \right)} = \frac{1}{2}$$
transform
$$- \frac{\cos{\left(2 x \right)}}{2} = 0$$
$$\sin^{2}{\left(x \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = - \frac{1}{2}$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-1/2) = 2
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{\sqrt{2}}{2}$$
Simplify
$$w_{2} = - \frac{\sqrt{2}}{2}$$
Simplify
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{4}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{4}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{5 \pi}{4}$$
Rapid solution
[src]
-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
pi
x2 = --
4
$$x_{2} = \frac{\pi}{4}$$
3*pi
x3 = ----
4
$$x_{3} = \frac{3 \pi}{4}$$
5*pi
x4 = ----
4
$$x_{4} = \frac{5 \pi}{4}$$
Sum and product of roots
[src]
sum
pi pi 3*pi 5*pi
0 - -- + -- + ---- + ----
4 4 4 4
$$\left(\left(\left(- \frac{\pi}{4} + 0\right) + \frac{\pi}{4}\right) + \frac{3 \pi}{4}\right) + \frac{5 \pi}{4}$$
=
2*pi
$$2 \pi$$
product
-pi pi 3*pi 5*pi 1*----*--*----*---- 4 4 4 4
$$\frac{5 \pi}{4} \frac{3 \pi}{4} \frac{\pi}{4} \cdot 1 \left(- \frac{\pi}{4}\right)$$
=
4 -15*pi ------- 256
$$- \frac{15 \pi^{4}}{256}$$
-15*pi^4/256
Numerical answer
[src]
x1 = 49.4800842940392
x2 = 162.577419823272
x3 = -35.3429173528852
x4 = 33.7721210260903
x5 = 63.6172512351933
x6 = -62.0464549083984
x7 = -85.6083998103219
x8 = -71.4712328691678
x9 = -90.3207887907066
x10 = -77.7544181763474
x11 = 8.63937979737193
x12 = -19.6349540849362
x13 = 87.1791961371168
x14 = -11.7809724509617
x15 = -41.6261026600648
x16 = -10.2101761241668
x17 = 82.4668071567321
x18 = 44.7676953136546
x19 = 32.2013246992954
x20 = 96.6039740978861
x21 = 60.4756585816035
x22 = -54.1924732744239
x23 = -18.0641577581413
x24 = -13.3517687777566
x25 = 47.9092879672443
x26 = -79.3252145031423
x27 = -24.3473430653209
x28 = -91.8915851175014
x29 = 5.49778714378214
x30 = -46.3384916404494
x31 = 30.6305283725005
x32 = -33.7721210260903
x33 = 2.35619449019234
x34 = 16.4933614313464
x35 = -49.4800842940392
x36 = 66.7588438887831
x37 = -1144.32512407008
x38 = -55.7632696012188
x39 = -3.92699081698724
x40 = 3.92699081698724
x41 = -84.037603483527
x42 = 84.037603483527
x43 = 85.6083998103219
x44 = -69.9004365423729
x45 = 54.1924732744239
x46 = -2.35619449019234
x47 = -57.3340659280137
x48 = -40.0553063332699
x49 = 22.776546738526
x50 = -68.329640215578
x51 = -76.1836218495525
x52 = 38.484510006475
x53 = 41.6261026600648
x54 = 88.7499924639117
x55 = 24.3473430653209
x56 = 27.4889357189107
x57 = 384.059701901352
x58 = 52.621676947629
x59 = -60.4756585816035
x60 = 91.8915851175014
x61 = -98.174770424681
x62 = -32.2013246992954
x63 = 19.6349540849362
x64 = 74.6128255227576
x65 = 69.9004365423729
x66 = -47.9092879672443
x67 = -63.6172512351933
x68 = -5.49778714378214
x69 = -99.7455667514759
x70 = -16.4933614313464
x71 = 99.7455667514759
x72 = 25.9181393921158
x73 = 98.174770424681
x74 = 76.1836218495525
x75 = 62.0464549083984
x76 = 55.7632696012188
x77 = -25.9181393921158
x78 = -27.4889357189107
x79 = -82.4668071567321
x80 = -93.4623814442964
x81 = -38.484510006475
x82 = 18.0641577581413
x83 = 68.329640215578
x84 = 90.3207887907066
x85 = 46.3384916404494
x86 = 10.2101761241668
x87 = 40.0553063332699
x88 = 77.7544181763474
x89 = 11.7809724509617
x89 = 11.7809724509617
The graph