Mister Exam
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- Equation:
-
Equation (x-1)(x+9)=8x
- Equation (49^sinx)^cosx=7^(sqrt3)^sinx
- Equation 3(tg^2(x)-1)*sqrt(-5cosx)=0
- Equation 5/8/x=-37/48
- Express {x} in terms of y where:
- -1*x-19*y=-10
- 5*x+5*y=20
- 4*x-16*y=6
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
-
Identical expressions
- (forty-nine ^sinx)^cosx= seven ^(sqrt3)^sinx
- (49 to the power of sinus of x) to the power of co sinus of e of x equally 7 to the power of ( square root of 3) to the power of sinus of x
- (forty minus nine to the power of sinus of x) to the power of co sinus of e of x equally seven to the power of ( square root of 3) to the power of sinus of x
- (49^sinx)^cosx=7^(√3)^sinx
- (49sinx)cosx=7(sqrt3)sinx
- 49sinxcosx=7sqrt3sinx
- 49^sinx^cosx=7^sqrt3^sinx
(49^sinx)^cosx=7^(sqrt3)^sinx equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
/ sin(x)\
cos(x) | ___ |
/ sin(x)\ \\/ 3 /
\49 / = 7
$$\left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} = 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}}$$
Detail solution
Given the equation
$$\left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} = 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}}$$
transform
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + \left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} - 1 = 0$$
$$- 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}} + \left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
$$- 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}} + \left(49^{\sin{\left(x \right)}}\right)^{w} - 1 = 0$$
or
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + \left(49^{\sin{\left(x \right)}}\right)^{w} - 1 = 0$$
Do replacement
$$v = \left(49^{\sin{\left(x \right)}}\right)^{w}$$
we get
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + v - 1 = 0$$
or
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + v - 1 = 0$$
do backward replacement
$$\left(49^{\sin{\left(x \right)}}\right)^{w} = v$$
or
$$w = \frac{\log{\left(v \right)}}{\log{\left(49^{\sin{\left(x \right)}} \right)}}$$
The final answer
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:
$$\left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} = 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}}$$
transform
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + \left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} - 1 = 0$$
$$- 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}} + \left(49^{\sin{\left(x \right)}}\right)^{\cos{\left(x \right)}} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
$$- 7^{\left(\sqrt{3}\right)^{\sin{\left(x \right)}}} + \left(49^{\sin{\left(x \right)}}\right)^{w} - 1 = 0$$
or
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + \left(49^{\sin{\left(x \right)}}\right)^{w} - 1 = 0$$
Do replacement
$$v = \left(49^{\sin{\left(x \right)}}\right)^{w}$$
we get
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + v - 1 = 0$$
or
$$- 7^{3^{\frac{\sin{\left(x \right)}}{2}}} + v - 1 = 0$$
do backward replacement
$$\left(49^{\sin{\left(x \right)}}\right)^{w} = v$$
or
$$w = \frac{\log{\left(v \right)}}{\log{\left(49^{\sin{\left(x \right)}} \right)}}$$
The final answer
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:
Numerical answer
[src]
x1 = 48.3770246924634
x2 = 35.8106540781042
x3 = -102.419422679847
x4 = -20.7380136865121
x5 = -39.5875696080508
x6 = 78.9923612256875
x7 = 3.59413753953242
x8 = 35.0100640754304
x9 = -64.7203108367692
x10 = -27.0211989936917
x11 = 29.5274687709246
x12 = -52.9545302250839
x13 = 22.4436934610712
x14 = -2.68904776764716
x15 = -33.3043843008713
x16 = -77.2866814511284
x17 = 53.8596199969691
x18 = 73.5097659211817
x19 = 66.4259906113283
x20 = -78.0872714538022
x21 = -404.012317424467
x22 = -27.8217889963655
x23 = 4.39472754220625
x24 = 41.2932493826099
x25 = -59.2377155322634
x26 = -8.97223307482675
x27 = -2050.20686790552
x28 = -84.3704567609818
x29 = 16.9610981565654
x30 = 91.5587318400466
x31 = -1.88845776497333
x32 = -21.5386036891859
x33 = 97.8419171472262
x34 = 412.001182506206
x35 = 92.3593218427205
x36 = -40.3881596107247
x37 = 16.1605081538916
x38 = -83.569866758308
x39 = 67.2265806140021
x40 = 9.87732284671201
x41 = -14.4548283793325
x42 = 79.7929512283613
x43 = 47.5764346897895
x44 = 72.7091759185079
x45 = -58.4371255295896
x46 = 86.0761365355409
x47 = -71.8040861466226
x48 = 54.6602099996429
x49 = 23.244283463745
x50 = -34.1049743035451
x51 = -15.2554183820063
x52 = -89.8530520654875
x53 = 98.6425071499001
x54 = -71.0034961439488
x55 = -8.17164307215292
x56 = 60.9433953068225
x57 = -45.8707549152304
x58 = 60.1428053041487
x59 = 10.6779128493858
x60 = -46.6713449179043
x61 = 28.7268787682508
x62 = -65.520900839443
x63 = 42.0938393852838
x64 = -96.1362373726671
x65 = 85.2755465328671
x66 = -90.6536420681614
x67 = -96.936827375341
x68 = -52.15394022241
x68 = -52.15394022241