Mister Exam
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Identical expressions
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-
Similar expressions
- sqrt(sin(x)+1)=0
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sqrt(sin(x)-1)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
transform
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$\sqrt{w - 1} = 0$$
so
$$w - 1 = 0$$
Move free summands (without w)
from left part to right part, we given:
$$w = 1$$
We get the answer: w = 1
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
transform
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
$$\sqrt{\sin{\left(x \right)} - 1} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$\sqrt{w - 1} = 0$$
so
$$w - 1 = 0$$
Move free summands (without w)
from left part to right part, we given:
$$w = 1$$
We get the answer: w = 1
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
Sum and product of roots
[src]
sum
pi -- 2
$$\frac{\pi}{2}$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
pi -- 2
$$\frac{\pi}{2}$$
=
pi -- 2
$$\frac{\pi}{2}$$
pi/2
Numerical answer
[src]
x1 = 7.85398163397448
x2 = -86.3937979737193
x3 = 58.1194640914112
x4 = -67.5442420521806
x5 = -4.71238898038469
x6 = -54.9778714378214
x7 = 83.2522053201295
x8 = -98.9601685880785
x9 = -29.845130209103
x10 = 26.7035375555132
x11 = -48.6946861306418
x12 = -180.641577581413
x13 = -17.2787595947439
x14 = 20.4203522483337
x15 = 76.96902001295
x16 = 39.2699081698724
x17 = 14.1371669411541
x18 = 39.2699081698724
x19 = 32.9867228626928
x20 = 89.5353906273091
x21 = -10.9955742875643
x22 = 70.6858347057703
x23 = 45.553093477052
x24 = 76.9690200129499
x25 = 32.9867228626928
x26 = -23.5619449019235
x27 = 83.2522053201295
x28 = 64.4026493985908
x29 = -36.1283155162826
x30 = -48.6946861306418
x31 = -92.6769832808989
x32 = -4.7123889803847
x33 = -10.9955742875643
x34 = 1.5707963267949
x35 = -73.8274273593601
x36 = -92.6769832808989
x37 = -54.9778714378214
x38 = 215.199096770901
x39 = 108.384946548848
x40 = -61.261056745001
x41 = -42.4115008234622
x42 = -80.1106126665397
x43 = 51.8362787842316
x44 = 95.8185759344887
x44 = 95.8185759344887