Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2*x^2-x-1=x^2-5*x-(-1-x^2)
-
Equation cos(x)=3
-
Equation sin(x)+cos(x)=-1
-
Equation sin(x)-cos(x)=1
- Express {x} in terms of y where:
- 17*x+16*y=7
- 18*x-14*y=-12
- -8*x-15*y=15
- 20*x+5*y=-3
- Sum of series:
- cos(x)^2
- Graphing y =:
-
cos(x)^2
- Limit of the function:
-
cos(x)^2
-
Identical expressions
- cos(x)^ two = one
- co sinus of e of (x) squared equally 1
- co sinus of e of (x) to the power of two equally one
- cos(x)2=1
- cosx2=1
- cos(x)²=1
- cos(x) to the power of 2=1
- cosx^2=1
-
Similar expressions
- cos(x)^(2)=3/4
- cos(x^2)=0
- 6*cos(x)^(2)+7*cos(x)-3=0
- sin(x)*cos(x^2-18^0)^(2)*log(x)=0
- cosx^2=1
cos(x)^2=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos^{2}{\left(x \right)} = 1$$
transform
$$- \sin^{2}{\left(x \right)} = 0$$
$$\cos^{2}{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = 1$$
$$b = 0$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 1$$
Simplify
$$w_{2} = -1$$
Simplify
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{2} = \pi n + \pi$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{4} = \pi n$$
$$\cos^{2}{\left(x \right)} = 1$$
transform
$$- \sin^{2}{\left(x \right)} = 0$$
$$\cos^{2}{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = 1$$
$$b = 0$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-1) = 4
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 1$$
Simplify
$$w_{2} = -1$$
Simplify
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{2} = \pi n + \pi$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{4} = \pi n$$
Sum and product of roots
[src]
sum
0 + 0 + pi + 2*pi
$$\left(\left(0 + 0\right) + \pi\right) + 2 \pi$$
=
3*pi
$$3 \pi$$
product
1*0*pi*2*pi
$$2 \pi 1 \cdot 0 \pi$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 87.9645943357576
x2 = 53.4070753627408
x3 = -47.123890151099
x4 = -21.9911485864515
x5 = 75.3982241944528
x6 = 28.2743338652012
x7 = 62.8318524523063
x8 = -15.7079632965264
x9 = 25.1327410188866
x10 = -25.132741632083
x11 = -53.4070752836338
x12 = 6.28318528425126
x13 = -91.106187201329
x14 = -43.9822971745789
x15 = -18.8495561207399
x16 = -65.9734457650176
x17 = -18.8495556944209
x18 = -106.814150357553
x19 = 47.123890018392
x20 = -47.1238900492539
x21 = -31.4159267051849
x22 = 9.42477859080277
x23 = 40.8407042560881
x24 = 34.5575190304759
x25 = -87.9645943587732
x26 = -34.5575189701076
x27 = -3.14159311568248
x28 = 53.4070756765307
x29 = -72.2566308741333
x30 = 18.8495556796107
x31 = -69.1150386737158
x32 = 84.8230014093114
x33 = 31.4159271479423
x34 = 62.8318528326557
x35 = -9.42477812668337
x36 = -12.5663700417108
x37 = 97.3893727097471
x38 = 84.8230010166547
x39 = -31.4159267959754
x40 = 25.1327414478072
x41 = 3.14159244884412
x42 = -50.2654822953391
x43 = 94.2477796093525
x44 = -78.5398160958028
x45 = 47.123889589354
x46 = -84.82300141007
x47 = -62.8318532583801
x48 = 56.5486676091327
x49 = 78.5398161878405
x50 = 50.2654824463473
x51 = -12.5663703661411
x52 = 72.256631027719
x53 = -25.132741473063
x54 = 15.7079634406648
x55 = 31.4159267865366
x56 = -34.5575189426108
x57 = 9.42477821024198
x58 = 69.1150381602162
x59 = 91.1061867314459
x60 = 40.840703919946
x61 = 97.3893725148693
x62 = -97.3893724403711
x63 = -62.8318528379059
x64 = 69.1150385885879
x65 = 43.982297169427
x66 = -100.530964672522
x67 = -59.6902604576401
x68 = 37.6991120192083
x69 = 65.9734457528975
x70 = 18.8495554002244
x71 = -1734.15914475848
x72 = 100.530964766599
x73 = 0.0
x74 = 59.6902605976901
x75 = -84.8230018263493
x76 = 12.5663704518704
x77 = -56.5486675191652
x78 = -37.6991118771514
x79 = -6.28318513794069
x80 = 3.14159287686128
x81 = -75.3982238620294
x82 = -40.8407042660168
x83 = 81.6814091761104
x84 = 21.9911485851964
x85 = -40.8407046898283
x86 = -94.2477794529919
x87 = 75.3982239388525
x88 = -69.1150386253436
x89 = -3.14159289677385
x90 = -81.6814090380061
x91 = -91.1061872003049
x92 = 91.1061871583643
x93 = -28.2743337166085
x93 = -28.2743337166085
The graph