Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 3*x=28-x
-
Equation 3(x-7)-6x=-x
-
Equation cos(x)^(2)=(1/2)
-
Equation (x+3)^2-16=(1-2x)^2
- Express {x} in terms of y where:
- 9*x-8*y=-6
- -18*x+16*y=4
- -9*x-19*y=13
- -13*x-5*y=6
- Integral of d{x}:
-
cos(x)^(2)
- Derivative of:
-
cos(x)^(2)
- (1/2)
- Graphing y =:
- cos(x)^(2)
- Sum of series:
-
(1/2)
-
Identical expressions
- cos(x)^(two)=(one / two)
- co sinus of e of (x) to the power of (2) equally (1 divide by 2)
- co sinus of e of (x) to the power of (two) equally (one divide by two)
- cos(x)(2)=(1/2)
- cosx2=1/2
- cosx^2=1/2
- cos(x)^(2)=(1 divide by 2)
-
Similar expressions
- (5/6*x-1/2)*5/6=5/6
- 1/2*x=5
- ((a-1)*x^2-a-1)/((2*a^2-2)*x^((a-1)/a))=y
- cosx^(2)=(1/2)
cos(x)^(2)=(1/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos^{2}{\left(x \right)} = \frac{1}{2}$$
transform
$$\frac{\cos{\left(2 x \right)}}{2} = 0$$
$$\cos^{2}{\left(x \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$w_{1} = \frac{\sqrt{2}}{2}$$
$$w_{2} = - \frac{\sqrt{2}}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = \pi n + \frac{3 \pi}{4}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{3} = \pi n - \frac{3 \pi}{4}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{4} = \pi n - \frac{\pi}{4}$$
$$\cos^{2}{\left(x \right)} = \frac{1}{2}$$
transform
$$\frac{\cos{\left(2 x \right)}}{2} = 0$$
$$\cos^{2}{\left(x \right)} - \frac{1}{2} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$w_{1} = \frac{\sqrt{2}}{2}$$
$$w_{2} = - \frac{\sqrt{2}}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = \pi n + \frac{3 \pi}{4}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{3} = \pi n - \frac{3 \pi}{4}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{4} = \pi n - \frac{\pi}{4}$$
Rapid solution
[src]
pi
x1 = --
4
$$x_{1} = \frac{\pi}{4}$$
3*pi
x2 = ----
4
$$x_{2} = \frac{3 \pi}{4}$$
5*pi
x3 = ----
4
$$x_{3} = \frac{5 \pi}{4}$$
7*pi
x4 = ----
4
$$x_{4} = \frac{7 \pi}{4}$$
x4 = 7*pi/4
Sum and product of roots
[src]
sum
pi 3*pi 5*pi 7*pi -- + ---- + ---- + ---- 4 4 4 4
$$\frac{7 \pi}{4} + \left(\left(\frac{\pi}{4} + \frac{3 \pi}{4}\right) + \frac{5 \pi}{4}\right)$$
=
4*pi
$$4 \pi$$
product
pi 3*pi 5*pi 7*pi --*----*----*---- 4 4 4 4
$$\frac{7 \pi}{4} \frac{5 \pi}{4} \frac{\pi}{4} \frac{3 \pi}{4}$$
=
4 105*pi ------- 256
$$\frac{105 \pi^{4}}{256}$$
105*pi^4/256
Numerical answer
[src]
x1 = 14247.9080821931
x2 = 32.2013246992954
x3 = 38.484510006475
x4 = 66.7588438887831
x5 = -1131.75875345572
x6 = -54.1924732744239
x7 = 5.49778714378214
x8 = 16.4933614313464
x9 = 54.1924732744239
x10 = 40.0553063332699
x11 = -69.9004365423729
x12 = -32.2013246992954
x13 = 41.6261026600648
x14 = -24.3473430653209
x15 = 82.4668071567321
x16 = 98.174770424681
x17 = -16.4933614313464
x18 = 77.7544181763474
x19 = 55.7632696012188
x20 = 88.7499924639117
x21 = -33.7721210260903
x22 = -3.92699081698724
x23 = 63.6172512351933
x24 = -57.3340659280137
x25 = 91.8915851175014
x26 = 27.4889357189107
x27 = 69.9004365423729
x28 = -49.4800842940392
x29 = -13.3517687777566
x30 = 76.1836218495525
x31 = 24.3473430653209
x32 = -85.6083998103219
x33 = 8.63937979737193
x34 = 3.92699081698724
x35 = 99.7455667514759
x36 = 162.577419823272
x37 = -99.7455667514759
x38 = -47.9092879672443
x39 = 44.7676953136546
x40 = -91.8915851175014
x41 = 84.037603483527
x42 = -2.35619449019234
x43 = 74.6128255227576
x44 = -82.4668071567321
x45 = -76.1836218495525
x46 = -19.6349540849362
x47 = 52.621676947629
x48 = -40.0553063332699
x49 = 85.6083998103219
x50 = -93.4623814442964
x51 = -27.4889357189107
x52 = -98.174770424681
x53 = -18.0641577581413
x54 = 33.7721210260903
x55 = -46.3384916404494
x56 = 30.6305283725005
x57 = -71.4712328691678
x58 = 11.7809724509617
x59 = 10.2101761241668
x60 = -41.6261026600648
x61 = 87.1791961371168
x62 = -68.329640215578
x63 = -25.9181393921158
x64 = 96.6039740978861
x65 = 25.9181393921158
x66 = -10.2101761241668
x67 = -84.037603483527
x68 = 18.0641577581413
x69 = -90.3207887907066
x70 = 22.776546738526
x71 = -38.484510006475
x72 = 47.9092879672443
x73 = 384.059701901352
x74 = 90.3207887907066
x75 = -60.4756585816035
x76 = -63.6172512351933
x77 = -79.3252145031423
x78 = 68.329640215578
x79 = -35.3429173528852
x80 = 49.4800842940392
x81 = 60.4756585816035
x82 = 2.35619449019234
x83 = -5.49778714378214
x84 = -12461.9126586273
x85 = 46.3384916404494
x86 = -77.7544181763474
x87 = -55.7632696012188
x88 = 62.0464549083984
x89 = 19.6349540849362
x90 = -62.0464549083984
x91 = -11.7809724509617
x91 = -11.7809724509617
The graph