Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 81*x^3+18*x^2+x=0
-
Equation (-x-4)*(3*x+3)=0
- Equation x^2-144=0
-
Equation -x=5
- Express {x} in terms of y where:
- -2*x-19*y=3
- 15*x+10*y=-4
- 15*x+9*y=5
- -9*x+5*y=-2
- Derivative of:
-
sin(x/2)
- -1/2
- Graphing y =:
- sin(x/2)
- Integral of d{x}:
-
sin(x/2)
-
-1/2
- Limit of the function:
- -1/2
-
Identical expressions
- sin(x/ two)=- one / two
- sinus of (x divide by 2) equally minus 1 divide by 2
- sinus of (x divide by two) equally minus one divide by two
- sinx/2=-1/2
- sin(x divide by 2)=-1 divide by 2
-
Similar expressions
- sin(x/2)=+1/2
- Sqrt(2(sin(x/2*(1-cos(x)))^2))=-sin(-x)-5cos(x)
- 3*(2*x+7)-2*(x-4)+4*x*x=-(1/2)
- 1/(4*(x-1))+1/4-1/(2x-1)=1/12
- 2*sin(x/2-pi/6)=1
- x=1+(sin(x-(1/2)))/2
- sinx/2=8
sin(x/2)=-1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n - \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{7 \pi}{3}$$
$$\sin{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n - \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{7 \pi}{3}$$
Rapid solution
[src]
-pi
x1 = ----
3
$$x_{1} = - \frac{\pi}{3}$$
7*pi
x2 = ----
3
$$x_{2} = \frac{7 \pi}{3}$$
x2 = 7*pi/3
Sum and product of roots
[src]
sum
pi 7*pi - -- + ---- 3 3
$$- \frac{\pi}{3} + \frac{7 \pi}{3}$$
=
2*pi
$$2 \pi$$
product
-pi 7*pi ----*---- 3 3
$$- \frac{\pi}{3} \frac{7 \pi}{3}$$
=
2 -7*pi ------ 9
$$- \frac{7 \pi^{2}}{9}$$
-7*pi^2/9
Numerical answer
[src]
x1 = 45.0294947014537
x2 = -17.8023583703422
x3 = -63.8790506229925
x4 = 49.2182849062401
x5 = 19.8967534727354
x6 = -13.6135681655558
x7 = -26.1799387799149
x8 = -30.3687289847013
x9 = 195.825942073764
x10 = 57.5958653158129
x11 = 74.3510261349584
x12 = -93.2005820564972
x13 = -5.23598775598299
x14 = -76.4454212373516
x15 = 32.4631240870945
x16 = -42.9350995990605
x17 = 36.6519142918809
x18 = 479288.705614517
x19 = -38.7463093942741
x20 = -51.3126800086333
x21 = -68.0678408277789
x22 = 61.7846555205993
x23 = 70.162235930172
x24 = -1.0471975511966
x25 = 99.4837673636768
x26 = 95.2949771588904
x27 = -55.5014702134197
x28 = 82.7286065445312
x29 = -80.634211442138
x30 = 7.33038285837618
x31 = -3762.58080144938
x32 = -101.57816246607
x33 = 11.5191730631626
x34 = 24.0855436775217
x35 = 86.9173967493176
x36 = -89.0117918517108
x36 = -89.0117918517108
The graph