Mister Exam
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Equation sin(x)=1/5
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- cosx/(one -sinx)= zero
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Similar expressions
- cosx/(1+sinx)=0
cosx/(1-sinx)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
cos(x) ---------- = 0 1 - sin(x)
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} = 0$$
Detail solution
Given the equation
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} = 0$$
transform
$$- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} - 1} = 0$$
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$\frac{\cos{\left(x \right)}}{1 - w} = 0$$
Multiply the equation sides by the denominator 1 - w
we get:
$$- \frac{\left(1 - w\right) \cos{\left(x \right)}}{w - 1} = 0$$
Expand brackets in the left part
Looking for similar summands in the left part:
Move free summands (without w)
from left part to right part, we given:
$$- \frac{\left(1 - w\right) \cos{\left(x \right)}}{w - 1} + 1 = 1$$
This equation has no roots
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:
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} = 0$$
transform
$$- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} - 1} = 0$$
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$\frac{\cos{\left(x \right)}}{1 - w} = 0$$
Multiply the equation sides by the denominator 1 - w
we get:
$$- \frac{\left(1 - w\right) \cos{\left(x \right)}}{w - 1} = 0$$
Expand brackets in the left part
-1-wcosx-1+w = 0
Looking for similar summands in the left part:
-(1 - w)*cos(x)/(-1 + w) = 0
Move free summands (without w)
from left part to right part, we given:
$$- \frac{\left(1 - w\right) \cos{\left(x \right)}}{w - 1} + 1 = 1$$
This equation has no roots
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:
Sum and product of roots
[src]
sum
3*pi ---- 2
$$\frac{3 \pi}{2}$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
product
3*pi ---- 2
$$\frac{3 \pi}{2}$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
3*pi/2
Numerical answer
[src]
x1 = 48.6946861306418
x2 = 54.9778714378214
x3 = 67.5442420521806
x4 = 36.1283155162826
x5 = 61.261056745001
x6 = -14.1371669411541
x7 = -64.4026493985908
x8 = 92.6769832808989
x9 = -51.8362787842316
x10 = 42.4115008234622
x11 = -45.553093477052
x12 = -89.5353906273091
x13 = -1.5707963267949
x14 = 23.5619449019235
x15 = -102.101761241668
x16 = -58.1194640914112
x17 = 73.8274273593601
x18 = 29.845130209103
x19 = 4.71238898038469
x20 = 86.3937979737193
x21 = -20.4203522483337
x22 = -26.7035375555132
x23 = 98.9601685880785
x24 = -95.8185759344887
x25 = 80.1106126665397
x26 = -7.85398163397448
x27 = -83.2522053201295
x28 = -76.9690200129499
x29 = -32.9867228626928
x30 = 17.2787595947439
x31 = -70.6858347057703
x32 = 10.9955742875643
x33 = -39.2699081698724
x33 = -39.2699081698724