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Similar expressions
- sin^2(x)+cos^2(x)=1
sin^2(x)-cos^2(x)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 2 sin (x) - cos (x) = 1
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} = 1$$
Detail solution
Given the equation
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} = 1$$
transform
$$- 2 \cos^{2}{\left(x \right)} = 0$$
$$2 \sin^{2}{\left(x \right)} - 2 = 0$$
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
$$a = 2$$
$$b = 0$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 1$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} = 1$$
transform
$$- 2 \cos^{2}{\left(x \right)} = 0$$
$$2 \sin^{2}{\left(x \right)} - 2 = 0$$
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
$$a = 2$$
$$b = 0$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (2) * (-2) = 16
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 1$$
$$w_{2} = -1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Sum and product of roots
[src]
sum
pi pi - -- + -- 2 2
$$- \frac{\pi}{2} + \frac{\pi}{2}$$
=
0
$$0$$
product
-pi pi ----*-- 2 2
$$- \frac{\pi}{2} \frac{\pi}{2}$$
=
2 -pi ----- 4
$$- \frac{\pi^{2}}{4}$$
-pi^2/4
Rapid solution
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
pi
x2 = --
2
$$x_{2} = \frac{\pi}{2}$$
x2 = pi/2
Numerical answer
[src]
x1 = 10.9955743696636
x2 = -7.85398149857354
x3 = -1.57079642969308
x4 = 45.553093700501
x5 = 29.845130320338
x6 = -73.8274272800405
x7 = 89.5353908552844
x8 = 541.924732890135
x9 = 80.1106131434937
x10 = -92.6769830239371
x11 = -14.1371668392726
x12 = -89.5353907467661
x13 = 48.6946859238715
x14 = 54.9778711883962
x15 = 80.1106126771746
x16 = -32.9867227513827
x17 = -45.5530935883361
x18 = -76.9690202568697
x19 = -4.71238872430683
x20 = 67.5442422779275
x21 = 92.6769830795146
x22 = -61.2610562242523
x23 = 61.2610566752601
x24 = -17.2787598091171
x25 = 76.9690207492347
x26 = -54.9778716831146
x27 = -4.7123889912442
x28 = 17.2787598502655
x29 = 64.4026493086922
x30 = 83.2522052340866
x31 = 20.4203521497111
x32 = 76.9690200400775
x33 = 83.2522055730903
x34 = 26.7035373461441
x35 = -70.6858346386357
x36 = -10.9955745350309
x37 = 39.2699081179815
x38 = -48.6946860920117
x39 = -29.8451300963672
x40 = -42.4115006098842
x41 = -39.2699083866483
x42 = -86.393797765473
x43 = -76.9690198771149
x44 = -83.2522055415057
x45 = 39.2699084246933
x46 = -17.2787590276524
x47 = 86.393797888273
x48 = 7.85398174058521
x49 = -20.4203520321877
x50 = 70.6858345016621
x51 = 17.2787595624179
x52 = -80.1106125795659
x53 = -26.7035372990183
x54 = 73.8274274795554
x55 = 54.9778714849733
x56 = 98.9601683381274
x57 = 23.5619449395428
x58 = -48.6946858738636
x59 = -70.685834448838
x60 = 23.5619451230057
x61 = -67.5442421675773
x62 = -39.2699081528781
x63 = 4.71238876848081
x64 = -23.5619450090417
x65 = -98.96016883042
x66 = -61.2610569641117
x67 = 14.1371671048484
x68 = -64.4026491876462
x69 = 36.1283156002139
x70 = -98.9601684414698
x71 = -36.1283154192437
x72 = 98.9601685932308
x73 = 32.9867226137576
x74 = -58.1194639993376
x75 = 1.5707965454425
x76 = -54.9778713137198
x77 = 51.8362788999928
x78 = 42.4115007291722
x79 = -26.7035375427973
x80 = 61.2610569989704
x81 = -51.8362786897497
x82 = -10.9955741902138
x83 = 10.9955740392793
x84 = 58.1194644379895
x85 = -98.960168684456
x86 = 76.9690197631883
x87 = 32.986722928111
x88 = -95.8185758681287
x89 = -32.9867231091652
x90 = -92.6769831823972
x91 = 95.8185760590309
x91 = 95.8185760590309