Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2=11
-
Equation sin(x^2)=0
-
Equation cos(x)=-3/4
-
Equation 13-6(3-x)=25
- Express {x} in terms of y where:
- -16*x+7*y=-5
- (x^2+y^2-1)^3-x^2*y^3=0
- -2*x+1*y=-8
- -2*x-1*y=-8
-
Identical expressions
- two sin^2(x)-5sinx- seven = zero
- 2 sinus of squared (x) minus 5 sinus of x minus 7 equally 0
- two sinus of squared (x) minus 5 sinus of x minus seven equally zero
- 2sin2(x)-5sinx-7=0
- 2sin2x-5sinx-7=0
- 2sin²(x)-5sinx-7=0
- 2sin to the power of 2(x)-5sinx-7=0
- 2sin^2x-5sinx-7=0
- 2sin^2(x)-5sinx-7=O
-
Similar expressions
- 2sin^2(x)-5sinx+7=0
- 2sin^2(x)+5sinx-7=0
2sin^2(x)-5sinx-7=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2*sin (x) - 5*sin(x) - 7 = 0
$$\left(2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 7 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 7 = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 7 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 7 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = 2$$
$$b = -5$$
$$c = -7$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{7}{2}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
$$\left(2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 7 = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)} - 7 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 7 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = 2$$
$$b = -5$$
$$c = -7$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (2) * (-7) = 81
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{7}{2}$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{7}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Rapid solution
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
3*pi
x2 = ----
2
$$x_{2} = \frac{3 \pi}{2}$$
x3 = pi - re(asin(7/2)) - I*im(asin(7/2))
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}$$
x4 = I*im(asin(7/2)) + re(asin(7/2))
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}$$
x4 = re(asin(7/2)) + i*im(asin(7/2))
Sum and product of roots
[src]
sum
pi 3*pi - -- + ---- + pi - re(asin(7/2)) - I*im(asin(7/2)) + I*im(asin(7/2)) + re(asin(7/2)) 2 2
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right) + \left(\left(- \frac{\pi}{2} + \frac{3 \pi}{2}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right)\right)$$
=
2*pi
$$2 \pi$$
product
-pi 3*pi ----*----*(pi - re(asin(7/2)) - I*im(asin(7/2)))*(I*im(asin(7/2)) + re(asin(7/2))) 2 2
$$- \frac{\pi}{2} \frac{3 \pi}{2} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right)$$
=
2
3*pi *(I*im(asin(7/2)) + re(asin(7/2)))*(-pi + I*im(asin(7/2)) + re(asin(7/2)))
-------------------------------------------------------------------------------
4
$$\frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right)}{4}$$
3*pi^2*(i*im(asin(7/2)) + re(asin(7/2)))*(-pi + i*im(asin(7/2)) + re(asin(7/2)))/4
Numerical answer
[src]
x1 = 98.9601682849072
x2 = -20.4203520172217
x3 = 4.71238875390585
x4 = -7.85398195102053
x5 = -83.2522050097403
x6 = 48.694685910795
x7 = -45.5530935898961
x8 = 36.1283157068514
x9 = -76.969019662615
x10 = 80.1106124827858
x11 = -64.402649662953
x12 = 48.6946864186186
x13 = 54.977871788106
x14 = -45900.2394649692
x15 = 86.3937978954211
x16 = -14.1371668462941
x17 = -58.1194639984133
x18 = 42.4115007866243
x19 = 29.8451303219183
x20 = 67.5442417600206
x21 = 86.3937978875539
x22 = -7.85398161800994
x23 = -1.57079632591833
x24 = 17.2787599147175
x25 = 23.5619451395792
x26 = -64.402649174177
x27 = -51.8362786895144
x28 = -14.137166838045
x29 = 10.9955746381689
x30 = -83.2522055594468
x31 = 10.9955739762081
x32 = 80.1106131397298
x33 = -7.85398149749143
x34 = -95.818575868085
x35 = -26.7035372393486
x36 = -89.5353907487017
x37 = -32.9867225120752
x38 = 61.2610570685864
x39 = -89.5353905609898
x40 = -2426.88032454877
x41 = -39.2699078662147
x42 = -39.2699084028422
x43 = -45.5530934418906
x44 = 29.845130261015
x45 = -32.9867231778458
x46 = 42.4115007282171
x47 = -7.85398202878013
x48 = 4.71238927659118
x49 = 61.2610564009947
x50 = -76.9690203316961
x51 = 98.9601689377237
x52 = -58.1194639494126
x53 = 23.5619446181122
x54 = -1.57079643092736
x55 = -1.57079615085824
x56 = 17.2787592507351
x57 = -70.6858350487702
x58 = 36.128316008479
x59 = -70.6858343936081
x60 = -20.4203525229045
x61 = 92.6769835598018
x62 = 92.6769830678231
x63 = 67.5442422962251
x64 = -26.7035378991915
x65 = 73.8274274815403
x66 = 54.9778711304608
x67 = 73.827427372192
x67 = 73.827427372192