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- Equation:
- Equation (7)/(1-cos^2(x))+(9)/(sin(x))=10
- Equation m-b/b2+m2(b+m/b-2b/b-m)
-
Equation 3x^2-4x=0
- Equation sin(x)=1/sqrt(2)
- Express {x} in terms of y where:
- 15*x-15*y=16
- 9*x+18*y=6
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- Sum of series:
-
10
- Derivative of:
-
10
- Integral of d{x}:
-
10
-
Identical expressions
- (seven)/(one -cos^ two (x))+(nine)/(sin(x))= ten
- (7) divide by (1 minus co sinus of e of squared (x)) plus (9) divide by ( sinus of (x)) equally 10
- (seven) divide by (one minus co sinus of e of to the power of two (x)) plus (nine) divide by ( sinus of (x)) equally ten
- (7)/(1-cos2(x))+(9)/(sin(x))=10
- 7/1-cos2x+9/sinx=10
- (7)/(1-cos²(x))+(9)/(sin(x))=10
- (7)/(1-cos to the power of 2(x))+(9)/(sin(x))=10
- 7/1-cos^2x+9/sinx=10
- (7) divide by (1-cos^2(x))+(9) divide by (sin(x))=10
-
Similar expressions
- (7)/(1-cos^2(x))-(9)/(sin(x))=10
- (7)/(1+cos^2(x))+(9)/(sin(x))=10
- (7)/(1-cos^2(x))+(9)/(sinx)=10
(7)/(1-cos^2(x))+(9)/(sin(x))=10 equation
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The solution
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7 9
----------- + ------ = 10
2 sin(x)
1 - cos (x)
$$\frac{9}{\sin{\left(x \right)}} + \frac{7}{1 - \cos^{2}{\left(x \right)}} = 10$$
Detail solution
Given the equation
$$\frac{9}{\sin{\left(x \right)}} + \frac{7}{1 - \cos^{2}{\left(x \right)}} = 10$$
transform
$$-10 + \frac{9}{\sin{\left(x \right)}} + \frac{7}{\sin^{2}{\left(x \right)}} = 0$$
$$-10 + \frac{9}{\sin{\left(x \right)}} + \frac{7}{\sin^{2}{\left(x \right)}} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$-10 + \frac{9}{w} + \frac{7}{w^{2}} = 0$$
Multiply the equation sides by the denominators:
w^2
we get:
$$w^{2} \left(-10 + \frac{9}{w} + \frac{7}{w^{2}}\right) = 0$$
$$- 10 w^{2} + 9 w + 7 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -10$$
$$b = 9$$
$$c = 7$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = \frac{7}{5}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$\frac{9}{\sin{\left(x \right)}} + \frac{7}{1 - \cos^{2}{\left(x \right)}} = 10$$
transform
$$-10 + \frac{9}{\sin{\left(x \right)}} + \frac{7}{\sin^{2}{\left(x \right)}} = 0$$
$$-10 + \frac{9}{\sin{\left(x \right)}} + \frac{7}{\sin^{2}{\left(x \right)}} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$-10 + \frac{9}{w} + \frac{7}{w^{2}} = 0$$
Multiply the equation sides by the denominators:
w^2
we get:
$$w^{2} \left(-10 + \frac{9}{w} + \frac{7}{w^{2}}\right) = 0$$
$$- 10 w^{2} + 9 w + 7 = 0$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -10$$
$$b = 9$$
$$c = 7$$
, then
D = b^2 - 4 * a * c =
(9)^2 - 4 * (-10) * (7) = 361
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = \frac{7}{5}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{5} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{5} \right)}$$
Rapid solution
[src]
-5*pi
x1 = -----
6
$$x_{1} = - \frac{5 \pi}{6}$$
-pi
x2 = ----
6
$$x_{2} = - \frac{\pi}{6}$$
/ / ___\\ / / ___\\
| |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||
x3 = 2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------||
\ \7 7 // \ \7 7 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)}$$
/ / ___\\ / / ___\\
| |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||
x4 = 2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------||
\ \7 7 // \ \7 7 //
$$x_{4} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)}$$
x4 = 2*re(atan(5/7 + 2*sqrt(6)*i/7)) + 2*i*im(atan(5/7 + 2*sqrt(6)*i/7))
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ 5*pi pi | |5 2*I*\/ 6 || | |5 2*I*\/ 6 || | |5 2*I*\/ 6 || | |5 2*I*\/ 6 || - ---- - -- + 2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------|| + 2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------|| 6 6 \ \7 7 // \ \7 7 // \ \7 7 // \ \7 7 //
$$\left(\left(- \frac{5 \pi}{6} - \frac{\pi}{6}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |5 2*I*\/ 6 || | |5 2*I*\/ 6 || | |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||
-pi + 2*re|atan|- - ---------|| + 2*re|atan|- + ---------|| + 2*I*im|atan|- - ---------|| + 2*I*im|atan|- + ---------||
\ \7 7 // \ \7 7 // \ \7 7 // \ \7 7 //
$$- \pi + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)}$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -5*pi -pi | | |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||| | | |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||| -----*----*|2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------|||*|2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------||| 6 6 \ \ \7 7 // \ \7 7 /// \ \ \7 7 // \ \7 7 ///
$$- \frac{5 \pi}{6} \left(- \frac{\pi}{6}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
2 | | |5 2*I*\/ 6 || | |5 2*I*\/ 6 ||| | | |5 2*I*\/ 6 || | |5 2*I*\/ 6 |||
5*pi *|I*im|atan|- - ---------|| + re|atan|- - ---------|||*|I*im|atan|- + ---------|| + re|atan|- + ---------|||
\ \ \7 7 // \ \7 7 /// \ \ \7 7 // \ \7 7 ///
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9
$$\frac{5 \pi^{2} \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} - \frac{2 \sqrt{6} i}{7} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{5}{7} + \frac{2 \sqrt{6} i}{7} \right)}\right)}\right)}{9}$$
5*pi^2*(i*im(atan(5/7 - 2*i*sqrt(6)/7)) + re(atan(5/7 - 2*i*sqrt(6)/7)))*(i*im(atan(5/7 + 2*i*sqrt(6)/7)) + re(atan(5/7 + 2*i*sqrt(6)/7)))/9
Numerical answer
[src]
x1 = 100.007366139275
x2 = -82.2050077689329
x3 = 68.5914396033772
x4 = 87.4409955249159
x5 = -71.733032256967
x6 = 41.3643032722656
x7 = 74.8746249105567
x8 = 12.0427718387609
x9 = -65.4498469497874
x10 = 22.5147473507269
x11 = 30.8923277602996
x12 = 81.1578102177363
x13 = -59.1666616426078
x14 = 62.3082542961976
x15 = 93.7241808320955
x16 = 47.6474885794452
x17 = 56.025068989018
x18 = 79.0634151153431
x19 = 91.6297857297023
x20 = 18.3259571459405
x21 = 66.497044500984
x22 = -75.9218224617533
x23 = -31.9395253114962
x24 = 28.7979326579064
x25 = 53.9306738866248
x26 = -6.80678408277789
x27 = -69.6386371545737
x28 = 60.2138591938044
x29 = -25.6563400043166
x30 = -408.930643742271
x31 = -1158.72409039904
x32 = -27.7507351067098
x33 = -34.0339204138894
x34 = -13.0899693899575
x35 = 3.66519142918809
x36 = -84.2994028713261
x37 = 449.771348238939
x38 = 16.2315620435473
x39 = -52.8834763354282
x40 = 97.9129710368819
x41 = -40.317105721069
x42 = -132.47049022637
x43 = 49.7418836818384
x44 = 35.081117965086
x45 = 24.60914245312
x46 = 5.75958653158129
x47 = -21.4675497995303
x48 = -63.3554518473942
x49 = -78.0162175641465
x50 = -38.2227106186758
x51 = 9.94837673636768
x52 = -57.0722665402146
x53 = -19.3731546971371
x53 = -19.3731546971371