Mister Exam
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How to use it?
- Equation:
- Equation x^2-4i=0
-
Equation 600-(x-37)=417
-
Equation 17-8y=4y-7
-
Equation 3x²-27=0
- Express {x} in terms of y where:
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
- 10*x-1*y=10
- 15*x-17*y=19
- 9*x-8*y=6
-
Identical expressions
- sin(fifty-seven)+sin(forty-one)= two *sin(x)*cos(eight)
- sinus of (57) plus sinus of (41) equally 2 multiply by sinus of (x) multiply by co sinus of e of (8)
- sinus of (fifty minus seven) plus sinus of (forty minus one) equally two multiply by sinus of (x) multiply by co sinus of e of (eight)
- sin(57)+sin(41)=2sin(x)cos(8)
- sin57+sin41=2sinxcos8
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Similar expressions
- sin(57)-sin(41)=2*sin(x)*cos(8)
- sin(57)+sin(41)=2*sinx*cos(8)
sin(57)+sin(41)=2*sin(x)*cos(8) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
sin(57) + sin(41) = 2*sin(x)*cos(8)
$$\sin{\left(41 \right)} + \sin{\left(57 \right)} = 2 \sin{\left(x \right)} \cos{\left(8 \right)}$$
Detail solution
Given the equation
$$\sin{\left(41 \right)} + \sin{\left(57 \right)} = 2 \sin{\left(x \right)} \cos{\left(8 \right)}$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi$$
, where n - is a integer
$$\sin{\left(41 \right)} + \sin{\left(57 \right)} = 2 \sin{\left(x \right)} \cos{\left(8 \right)}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2*cos(8)
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{- \sin{\left(57 \right)} - \sin{\left(41 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi$$
, where n - is a integer
Rapid solution
[src]
/sin(41) + sin(57)\
x_1 = pi - asin|-----------------|
\ 2*cos(8) /
$$x_{1} = - \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi$$
/sin(41) + sin(57)\
x_2 = asin|-----------------|
\ 2*cos(8) /
$$x_{2} = \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
Sum and product of roots
[src]
sum
/sin(41) + sin(57)\ /sin(41) + sin(57)\
pi - asin|-----------------| + asin|-----------------|
\ 2*cos(8) / \ 2*cos(8) /
$$\left(- \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi\right) + \left(\operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)}\right)$$
=
pi
$$\pi$$
product
/sin(41) + sin(57)\ /sin(41) + sin(57)\
pi - asin|-----------------| * asin|-----------------|
\ 2*cos(8) / \ 2*cos(8) /
$$\left(- \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi\right) * \left(\operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)}\right)$$
=
/ /sin(41) + sin(57)\\ /sin(41) + sin(57)\ |pi - asin|-----------------||*asin|-----------------| \ \ 2*cos(8) // \ 2*cos(8) /
$$\left(- \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)} + \pi\right) \operatorname{asin}{\left(\frac{\sin{\left(41 \right)} + \sin{\left(57 \right)}}{2 \cos{\left(8 \right)}} \right)}$$
Numerical answer
[src]
x1 = 73.5221134900019
x2 = 55.2831853071796
x3 = 98.6548547187203
x4 = 99.2654824574367
x5 = -165.238928182822
x6 = -26.398223686155
x7 = 86.0884841043611
x8 = 1537.50429006285
x9 = -38.9645943005142
x10 = -146.389372261284
x11 = -39.5752220392306
x12 = 54.6725575684632
x13 = 10.6902604182061
x14 = 4.40707511102648
x15 = -70.3805208364121
x16 = -89.2300767579509
x17 = -89.8407044966673
x18 = 86.6991118430775
x19 = 49.0
x20 = 67.2389281828223
x21 = 3385.37139811236
x22 = -27.0088514248714
x23 = 23.2566310325652
x24 = 5.01770284974289
x25 = 92.9822971502571
x26 = -64.707963267949
x27 = -76.6637061435917
x28 = -14.4424808105123
x29 = 35.8230016469244
x30 = 67.8495559215388
x31 = 36.4336293856408
x32 = -20.1150383789755
x33 = -95.5132620651305
x34 = 60.9557428756428
x35 = -7.54866776461628
x36 = 74.1327412287183
x37 = -32.6814089933346
x38 = -96.1238898038469
x39 = 42.106186954104
x40 = -101.79644737231
x41 = -58.4247779607694
x42 = -1.26548245743669
x43 = -70.9911485751285
x44 = 124.398223686155
x45 = -82.9468914507713
x46 = -13.8318530717959
x47 = -64.0973355292326
x48 = 48.3893722612836
x49 = -1.8761101961531
x50 = -8.15929550333269
x51 = 23.8672587712817
x52 = 117.504410640259
x53 = -52.1415926535898
x54 = -51.5309649148734
x55 = -45.2477796076938
x56 = 30.1504440784612
x57 = -114.362817986669
x58 = -57.814150222053
x59 = 61.5663706143592
x60 = -77.2743338823081
x61 = -33.292036732051
x62 = 79.8052987971815
x63 = 80.4159265358979
x64 = -20.7256661176919
x65 = 17.5840734641021
x66 = 16.9734457253857
x67 = 92.3716694115407
x68 = 212.362817986669
x69 = 29.5398163397448
x70 = -45.8584073464102
x71 = 42.7168146928204
x72 = -83.5575191894877
x73 = 11.3008881569225
x73 = 11.3008881569225
The graph