Mister Exam
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How to use it?
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Equation sqrt(2*cos(x))-1=0
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- sqrt(2*cos(x))-1
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Identical expressions
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Similar expressions
- sqrt2*cos(x)-1=0
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sqrt(2*cos(x))-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{2 \cos{\left(x \right)}} - 1 = 0$$
transform
$$\sqrt{2} \sqrt{\cos{\left(x \right)}} - 1 = 0$$
$$\left(\sqrt{2 \cos{\left(x \right)}} - 1\right) + 0 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\sqrt{2} \sqrt{w} - 1 = 0$$
Because equation degree is equal to = 1/2 - does not contain even numbers in the numerator, then
the equation has single real root.
We raise the equation sides to 2-th degree:
We get:
$$\left(\sqrt{2 w + 0}\right)^{2} = 1^{2}$$
or
$$2 w = 1$$
Divide both parts of the equation by 2
We get the answer: w = 1/2
The final answer:
$$w_{1} = \frac{1}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
$$\sqrt{2 \cos{\left(x \right)}} - 1 = 0$$
transform
$$\sqrt{2} \sqrt{\cos{\left(x \right)}} - 1 = 0$$
$$\left(\sqrt{2 \cos{\left(x \right)}} - 1\right) + 0 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\sqrt{2} \sqrt{w} - 1 = 0$$
Because equation degree is equal to = 1/2 - does not contain even numbers in the numerator, then
the equation has single real root.
We raise the equation sides to 2-th degree:
We get:
$$\left(\sqrt{2 w + 0}\right)^{2} = 1^{2}$$
or
$$2 w = 1$$
Divide both parts of the equation by 2
w = 1 / (2)
We get the answer: w = 1/2
The final answer:
$$w_{1} = \frac{1}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = 2 \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 3 3
$$\left(\frac{\pi}{3}\right) + \left(\frac{5 \pi}{3}\right)$$
=
2*pi
$$2 \pi$$
product
pi 5*pi -- * ---- 3 3
$$\left(\frac{\pi}{3}\right) * \left(\frac{5 \pi}{3}\right)$$
=
2 5*pi ----- 9
$$\frac{5 \pi^{2}}{9}$$
Rapid solution
[src]
pi
x_1 = --
3
$$x_{1} = \frac{\pi}{3}$$
5*pi
x_2 = ----
3
$$x_{2} = \frac{5 \pi}{3}$$
Numerical answer
[src]
x1 = -76.4454212373516
x2 = 76.4454212373516
x3 = -61.7846555205993
x4 = 11.5191730631626
x5 = -86.9173967493176
x6 = -51.3126800086333
x7 = -26.1799387799149
x8 = 70.162235930172
x9 = 68.0678408277789
x10 = -57.5958653158129
x11 = 45.0294947014537
x12 = -55.5014702134197
x13 = -68.0678408277789
x14 = 19.8967534727354
x15 = -93.2005820564972
x16 = 32.4631240870945
x17 = 49.2182849062401
x18 = 74.3510261349584
x19 = -7.33038285837618
x20 = -70.162235930172
x21 = 5.23598775598299
x22 = -42.9350995990605
x23 = 17.8023583703422
x24 = -11.5191730631626
x25 = -45.0294947014537
x26 = 7.33038285837618
x27 = -30.3687289847013
x28 = 63.8790506229925
x29 = -74.3510261349584
x30 = -99.4837673636768
x31 = 26.1799387799149
x32 = -89.0117918517108
x33 = 95.2949771588904
x34 = 24.0855436775217
x35 = -13.6135681655558
x36 = -24.0855436775217
x37 = 51.3126800086333
x38 = -49.2182849062401
x39 = 57.5958653158129
x40 = 61.7846555205993
x41 = 38.7463093942741
x42 = 55.5014702134197
x43 = 99.4837673636768
x44 = -5.23598775598299
x45 = -63.8790506229925
x46 = 82.7286065445312
x47 = -1.0471975511966
x48 = -32.4631240870945
x49 = -82.7286065445312
x50 = -19.8967534727354
x51 = -36.6519142918809
x52 = 89.0117918517108
x53 = -80.634211442138
x54 = 80.634211442138
x55 = -9275.02871094827
x56 = 1.0471975511966
x57 = 86.9173967493176
x58 = -38.7463093942741
x59 = 36.6519142918809
x60 = 13.6135681655558
x61 = 42.9350995990605
x62 = 30.3687289847013
x63 = 93.2005820564972
x64 = -95.2949771588904
x65 = -17.8023583703422
x65 = -17.8023583703422
The graph