Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation sin^2x=1/2
- Equation (x^2-16)^2+(x^2-3x-28)^2=0
-
Equation cos(x+pi/3)=(-1)/sqrt(2)
-
Equation x^4+16=0
- Express {x} in terms of y where:
- 5*x-5*y=20
- 8*x-7*y=-13
- 9*x+18*y=6
- 15*x-15*y=16
- Derivative of:
-
sin^2x
-
1/2
- Integral of d{x}:
-
sin^2x
-
1/2
- Graphing y =:
- sin^2x
- Sum of series:
-
1/2
-
Identical expressions
- sin^ two x= one /2
- sinus of squared x equally 1 divide by 2
- sinus of to the power of two x equally one divide by 2
- sin2x=1/2
- sin²x=1/2
- sin to the power of 2x=1/2
- sin^2x=1 divide by 2
-
Similar expressions
- 4sin^2x+9cosx-6=0
- 5*sin^2x+21*sinx+4=0
- sin^2((x/4)+(pi/4))*sin^2((x/4)-(pi/4))=0,375sin^2(-(pi/4))
sin^2x=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
$$\sin^{2}{\left(x \right)} - \frac{1}{2} = 0$$
$$- \frac{\cos{\left(2 x \right)}}{2} = 0$$
Consider each factor separately
$$\cos{\left(2 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(2 x \right)} = 0$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{acos}{\left(0 \right)}$$
$$2 x = 2 \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$2 x = 2 \pi n + \frac{\pi}{2}$$
$$2 x = 2 \pi n - \frac{\pi}{2}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{4}$$
$$x = \pi n - \frac{\pi}{4}$$
The final answer:
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
$$\sin^{2}{\left(x \right)} = \frac{1}{2}$$
Transform
$$\sin^{2}{\left(x \right)} - \frac{1}{2} = 0$$
$$- \frac{\cos{\left(2 x \right)}}{2} = 0$$
Consider each factor separately
Step
$$\cos{\left(2 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(2 x \right)} = 0$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{acos}{\left(0 \right)}$$
$$2 x = 2 \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$2 x = 2 \pi n + \frac{\pi}{2}$$
$$2 x = 2 \pi n - \frac{\pi}{2}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = \pi n + \frac{\pi}{4}$$
$$x = \pi n - \frac{\pi}{4}$$
The final answer:
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
Sum and product of roots
[src]
sum
-pi pi 3*pi 5*pi ---- + -- + ---- + ---- 4 4 4 4
$$\left(- \frac{\pi}{4}\right) + \left(\frac{\pi}{4}\right) + \left(\frac{3 \pi}{4}\right) + \left(\frac{5 \pi}{4}\right)$$
=
2*pi
$$2 \pi$$
product
-pi pi 3*pi 5*pi ---- * -- * ---- * ---- 4 4 4 4
$$\left(- \frac{\pi}{4}\right) * \left(\frac{\pi}{4}\right) * \left(\frac{3 \pi}{4}\right) * \left(\frac{5 \pi}{4}\right)$$
=
4 -15*pi ------- 256
$$- \frac{15 \pi^{4}}{256}$$
Rapid solution
[src]
-pi
x_1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
pi
x_2 = --
4
$$x_{2} = \frac{\pi}{4}$$
3*pi
x_3 = ----
4
$$x_{3} = \frac{3 \pi}{4}$$
5*pi
x_4 = ----
4
$$x_{4} = \frac{5 \pi}{4}$$
Numerical answer
[src]
x1 = -90.3207887907066
x2 = -85.6083998103219
x3 = -24.3473430653209
x4 = -35.3429173528852
x5 = -93.4623814442964
x6 = -71.4712328691678
x7 = -5.49778714378214
x8 = -84.037603483527
x9 = 44.7676953136546
x10 = 52.621676947629
x11 = 55.7632696012188
x12 = 46.3384916404494
x13 = 10.2101761241668
x14 = -1144.32512407008
x15 = 22.776546738526
x16 = -54.1924732744239
x17 = -99.7455667514759
x18 = -25.9181393921158
x19 = 2.35619449019234
x20 = -32.2013246992954
x21 = -11.7809724509617
x22 = 82.4668071567321
x23 = 98.174770424681
x24 = 33.7721210260903
x25 = 25.9181393921158
x26 = 8.63937979737193
x27 = -16.4933614313464
x28 = 66.7588438887831
x29 = -3.92699081698724
x30 = -62.0464549083984
x31 = -27.4889357189107
x32 = -77.7544181763474
x33 = -79.3252145031423
x34 = -57.3340659280137
x35 = 76.1836218495525
x36 = 99.7455667514759
x37 = 77.7544181763474
x38 = 96.6039740978861
x39 = 16.4933614313464
x40 = -60.4756585816035
x41 = 62.0464549083984
x42 = 69.9004365423729
x43 = 30.6305283725005
x44 = 87.1791961371168
x45 = 40.0553063332699
x46 = 384.059701901352
x47 = 68.329640215578
x48 = -19.6349540849362
x49 = 90.3207887907066
x50 = -38.484510006475
x51 = 85.6083998103219
x52 = -46.3384916404494
x53 = -47.9092879672443
x54 = 11.7809724509617
x55 = -63.6172512351933
x56 = 47.9092879672443
x57 = 41.6261026600648
x58 = 74.6128255227576
x59 = 49.4800842940392
x60 = -41.6261026600648
x61 = 3.92699081698724
x62 = -76.1836218495525
x63 = 5.49778714378214
x64 = 162.577419823272
x65 = 60.4756585816035
x66 = -13.3517687777566
x67 = 32.2013246992954
x68 = 84.037603483527
x69 = -82.4668071567321
x70 = 91.8915851175014
x71 = 19.6349540849362
x72 = -98.174770424681
x73 = -49.4800842940392
x74 = 88.7499924639117
x75 = 24.3473430653209
x76 = 18.0641577581413
x77 = -55.7632696012188
x78 = -68.329640215578
x79 = -10.2101761241668
x80 = -91.8915851175014
x81 = 27.4889357189107
x82 = -40.0553063332699
x83 = 63.6172512351933
x84 = -33.7721210260903
x85 = -69.9004365423729
x86 = 38.484510006475
x87 = 54.1924732744239
x88 = -18.0641577581413
x89 = -2.35619449019234
x89 = -2.35619449019234
The graph