Mister Exam

Other calculators

sin(x/2)=1/2

sin(x/2)=1/2 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src] 
   /x\      
sin|-| = 1/2
   \2/
sin⁡(x2)=12\sin{\left(\frac{x}{2} \right)} = \frac{1}{2}
Simplify
Detail solution
Given the equation
sin⁡(x2)=12\sin{\left(\frac{x}{2} \right)} = \frac{1}{2}
- this is the simplest trigonometric equation
This equation is transformed to
x2=2πn+asin⁡(12)\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}
x2=2πn−asin⁡(12)+π\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi
Or
x2=2πn+π6\frac{x}{2} = 2 \pi n + \frac{\pi}{6}
x2=2πn+5π6\frac{x}{2} = 2 \pi n + \frac{5 \pi}{6}
, where n - is a integer
Divide both parts of the equation by
12\frac{1}{2}
we get the answer:
x1=4πn+π3x_{1} = 4 \pi n + \frac{\pi}{3}
x2=4πn+5π3x_{2} = 4 \pi n + \frac{5 \pi}{3}
The graph
0-80-60-40-2020406080-1001002-2
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
     pi
x1 = --
     3
x1=π3x_{1} = \frac{\pi}{3} 
     5*pi
x2 = ----
      3
x2=5π3x_{2} = \frac{5 \pi}{3} 
x2 = 5*pi/3
Sum and product of roots [src] 
sum
pi   5*pi
-- + ----
3     3
π3+5π3\frac{\pi}{3} + \frac{5 \pi}{3} 
=
2*pi
2π2 \pi 
product
pi 5*pi
--*----
3   3
π35π3\frac{\pi}{3} \frac{5 \pi}{3} 
=
    2
5*pi 
-----
  9
5π29\frac{5 \pi^{2}}{9} 
5*pi^2/9
Numerical answer [src] 
x1 = 13.6135681655558
x2 = 5.23598775598299
x3 = -7.33038285837618
x4 = -74.3510261349584
x5 = -82.7286065445312
x6 = 101.57816246607
x7 = -70.162235930172
x8 = 93.2005820564972
x9 = -19.8967534727354
x10 = 42.9350995990605
x11 = -32.4631240870945
x12 = 55.5014702134197
x13 = -45.0294947014537
x14 = 30.3687289847013
x15 = -11.5191730631626
x16 = -61.7846555205993
x17 = 59100.6881968824
x18 = -99.4837673636768
x19 = -36.6519142918809
x20 = -24.0855436775217
x21 = 38.7463093942741
x22 = 1.0471975511966
x23 = 331.961623729322
x24 = -57.5958653158129
x25 = 68.0678408277789
x26 = -49.2182849062401
x27 = 26.1799387799149
x28 = 63.8790506229925
x29 = 114.144533080429
x30 = -86.9173967493176
x31 = -346.622389446074
x32 = -95.2949771588904
x33 = 17.8023583703422
x34 = 51.3126800086333
x35 = 76.4454212373516
x36 = 105.766952670856
x37 = 80.634211442138
x38 = 89.0117918517108
x38 = 89.0117918517108
The graph
sin(x/2)=1/2 equation
    © Mister Exam – Calculator