Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation sin(x)=3
-
Equation 2*sin(x)-1=0
-
Equation x^2-6x+5=0
-
Equation x^2=-49
- Express {x} in terms of y where:
- -18*x+9*y=-15
- 15*x-3*y=-8
- -2*x-19*y=3
- 4*x-7*y=12
- Derivative of:
-
sin(3*x-pi/4)
-
Identical expressions
- sin(three *x-pi/ four)=-sqrt(three)/ two
- sinus of (3 multiply by x minus Pi divide by 4) equally minus square root of (3) divide by 2
- sinus of (three multiply by x minus Pi divide by four) equally minus square root of (three) divide by two
- sin(3*x-pi/4)=-√(3)/2
- sin(3x-pi/4)=-sqrt(3)/2
- sin3x-pi/4=-sqrt3/2
- sin(3*x-pi divide by 4)=-sqrt(3) divide by 2
-
Similar expressions
- sin(3*x+pi/4)=-sqrt(3)/2
- sin(x)*cos(x)=(-sqrt(3))/((2*sin(x)))
- sin(3*x-pi/4)=+sqrt(3)/2
- cos(p/3+x)=(-sqrt(3))/2
- sin(2*x-pi/3)=-(sqrt(3)/2)
- sin(3*x-pi/6)=(-sqrt(3))/2
sin(3*x-pi/4)=-sqrt(3)/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
___ / pi\ -\/ 3 sin|3*x - --| = ------- \ 4 / 2
$$\sin{\left(3 x - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Detail solution
Given the equation
$$\sin{\left(3 x - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\cos{\left(3 x + \frac{\pi}{4} \right)} = \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
Or
$$3 x + \frac{\pi}{4} = \pi n + \frac{\pi}{6}$$
$$3 x + \frac{\pi}{4} = \pi n - \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n - \frac{\pi}{12}$$
$$3 x = \pi n - \frac{13 \pi}{12}$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} - \frac{13 \pi}{36}$$
$$\sin{\left(3 x - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -1
The equation is transformed to
$$\cos{\left(3 x + \frac{\pi}{4} \right)} = \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
Or
$$3 x + \frac{\pi}{4} = \pi n + \frac{\pi}{6}$$
$$3 x + \frac{\pi}{4} = \pi n - \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n - \frac{\pi}{12}$$
$$3 x = \pi n - \frac{13 \pi}{12}$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} - \frac{13 \pi}{36}$$
Rapid solution
[src]
-pi
x1 = ----
36
$$x_{1} = - \frac{\pi}{36}$$
19*pi
x2 = -----
36
$$x_{2} = \frac{19 \pi}{36}$$
x2 = 19*pi/36
Sum and product of roots
[src]
sum
pi 19*pi - -- + ----- 36 36
$$- \frac{\pi}{36} + \frac{19 \pi}{36}$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
-pi 19*pi ----*----- 36 36
$$- \frac{\pi}{36} \frac{19 \pi}{36}$$
=
2 -19*pi ------- 1296
$$- \frac{19 \pi^{2}}{1296}$$
-19*pi^2/1296
Numerical answer
[src]
x1 = 24.6964089157198
x2 = -88.0518607631139
x3 = -46.16395871525
x4 = -2.18166156499291
x5 = -63.2681853847944
x6 = 27.1398698685118
x7 = 20.8566845613322
x8 = -60.8247244320024
x9 = -35.691983203284
x10 = 75.3109572235553
x11 = 22.6020138133266
x12 = 35.5174502780846
x13 = -92.2406509679003
x14 = 52.2726110972302
x15 = -71.6457657943672
x16 = -6.3704517697793
x17 = -77.9289511015468
x18 = -50.7018147704353
x19 = 2.00712863979348
x20 = 16.318828506147
x21 = 18.4132236085402
x22 = -23.4746784393237
x23 = -10.9083078249646
x24 = -8.4648468721725
x25 = -41.9751685104636
x26 = 41.8006355852642
x27 = -83.8630705583275
x28 = 33.0739893252925
x29 = 64.4899158611905
x30 = 45.9894257900506
x31 = 98.0002374994816
x32 = 96.2549082474873
x33 = 87.8773278379145
x34 = -33.9466539512897
x35 = 89.9717229403077
x36 = 58.2067305540109
x37 = 92.0661180427009
x38 = -54.8906049752217
x39 = 81.5941425307349
x40 = 56.1123354516177
x41 = 66.5843109635837
x42 = -37.7863783056772
x43 = -98.8729021254788
x44 = -39.8807734080704
x45 = 100.094632601875
x46 = 4.10152374218667
x47 = -44.0695636128568
x48 = -85.9574656607207
x49 = 54.0179403492245
x50 = 43.8950306876574
x51 = -67.4569755895808
x52 = -4.27605666738611
x53 = -31.8522588488965
x54 = 85.7829327355213
x55 = 39.706240482871
x56 = 62.3955207587973
x57 = -108.995811787046
x58 = -96.4294411726867
x59 = 48.0838208924438
x60 = -48.2583538176432
x61 = -16.8424272817453
x62 = -69.551370691974
x63 = -73.7401608967604
x64 = 68.6787060659769
x65 = -0.0872664625997165
x66 = 83.6885376331281
x67 = -75.8345559991536
x68 = 10.0356431989674
x69 = 25.0454747661186
x70 = 6.19591884457987
x71 = -19.2858882345373
x72 = -80.02334620394
x73 = 50.178215994837
x74 = -29.7578637465033
x75 = -21.3802833369305
x76 = 8.29031394697306
x77 = -25.5690735417169
x78 = 60.3011256564041
x79 = -52.4471440224296
x80 = -94.3350460702935
x81 = 94.1605131450941
x82 = 20.5076187109334
x83 = 77.0562864755497
x84 = -81.7686754559343
x85 = 14.2244334037538
x86 = 41.4515697348653
x87 = 12.1300383013606
x88 = -65.3625804871876
x89 = -90.1462558655071
x90 = 71.1221670187689
x91 = -27.6634686441101
x92 = -50.3527489200364
x93 = 37.6118453804778
x94 = 79.4997474283417
x94 = 79.4997474283417