Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2+2x-3=0
-
Equation x-6/5=-1
-
Equation 1-2(5+3x)=15
-
Equation x^2-6x+9=0
- Express {x} in terms of y where:
- 2*x+2*y=-6
- 12*x-13*y=-14
- 1*x+5*y=-11
- -11*x-15*y=14
- Graphing y =:
- 2*cos(x)-1
-
Identical expressions
- two *cos(x)- one = zero
- 2 multiply by co sinus of e of (x) minus 1 equally 0
- two multiply by co sinus of e of (x) minus one equally zero
- 2cos(x)-1=0
- 2cosx-1=0
- 2*cos(x)-1=O
-
Similar expressions
- 2*cos(x)+1=0
- √(9-x^2)*(2cosx-1)=0
- 3^-2/2*cosx-1/2*sinx=1
- 2*cosx-1=0
2*cos(x)-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 \cos{\left(x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \cos{\left(x \right)} - 1 + 1 = 1$$
The equation is transformed to
$$\cos{\left(x \right)} = \frac{1}{2}$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$x = \pi n + \frac{\pi}{3}$$
$$x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
$$2 \cos{\left(x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move -1 to right part of the equation
with the change of sign in -1
We get:
$$2 \cos{\left(x \right)} - 1 + 1 = 1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(x \right)} = \frac{1}{2}$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$x = \pi n + \frac{\pi}{3}$$
$$x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Rapid solution
[src]
pi
x1 = --
3
$$x_{1} = \frac{\pi}{3}$$
5*pi
x2 = ----
3
$$x_{2} = \frac{5 \pi}{3}$$
Sum and product of roots
[src]
sum
pi 5*pi
0 + -- + ----
3 3
$$\left(0 + \frac{\pi}{3}\right) + \frac{5 \pi}{3}$$
=
2*pi
$$2 \pi$$
product
pi 5*pi 1*--*---- 3 3
$$\frac{5 \pi}{3} \cdot 1 \frac{\pi}{3}$$
=
2 5*pi ----- 9
$$\frac{5 \pi^{2}}{9}$$
5*pi^2/9
Numerical answer
[src]
x1 = -19.8967534727354
x2 = 24.0855436775217
x3 = -95.2949771588904
x4 = -89.0117918517108
x5 = 49.2182849062401
x6 = 38.7463093942741
x7 = 86.9173967493176
x8 = -45.0294947014537
x9 = -63.8790506229925
x10 = 1651.43053823704
x11 = 70.162235930172
x12 = -32.4631240870945
x13 = 5.23598775598299
x14 = -76.4454212373516
x15 = -36.6519142918809
x16 = 74.3510261349584
x17 = -86.9173967493176
x18 = -38.7463093942741
x19 = 42.9350995990605
x20 = -70.162235930172
x21 = -61.7846555205993
x22 = -7.33038285837618
x23 = 1.0471975511966
x24 = 51.3126800086333
x25 = 68.0678408277789
x26 = 82.7286065445312
x27 = -49.2182849062401
x28 = 76.4454212373516
x29 = -30.3687289847013
x30 = 13.6135681655558
x31 = 19.8967534727354
x32 = 80.634211442138
x33 = -17.8023583703422
x34 = -55.5014702134197
x35 = -51.3126800086333
x36 = 32.4631240870945
x37 = 55.5014702134197
x38 = -80.634211442138
x39 = 93.2005820564972
x40 = 95.2949771588904
x41 = 26.1799387799149
x42 = -225.147473507269
x43 = 30.3687289847013
x44 = 89.0117918517108
x45 = 11.5191730631626
x46 = 57.5958653158129
x47 = -68.0678408277789
x48 = 45.0294947014537
x49 = -74.3510261349584
x50 = -24.0855436775217
x51 = -13.6135681655558
x52 = 36.6519142918809
x53 = -5.23598775598299
x54 = 61.7846555205993
x55 = 63.8790506229925
x56 = 17.8023583703422
x57 = -11.5191730631626
x58 = -26.1799387799149
x59 = -82.7286065445312
x60 = -42.9350995990605
x61 = -99.4837673636768
x62 = -57.5958653158129
x63 = -359.188760060433
x64 = -1.0471975511966
x65 = 99.4837673636768
x66 = 7.33038285837618
x67 = -93.2005820564972
x67 = -93.2005820564972
The graph