Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
 

heureka

avatar
Benutzernameheureka
Punkte26396
Membership
Stats
Fragen 17
Antworten 5678

 #2
avatar+26396 
+4

For positive real numbers and the equation
arctanx+arccosy1+y2=arcsin310
reduces to an equation of the form xy+ax+by+c=0.

 

My attempt:

arctanx+arccosy1+y2=arcsin310|tan() both sidestan(arctanx+arccosy1+y2)=tan(arcsin310)sin(arctanx+arccosy1+y2)cos(arctanx+arccosy1+y2)=sin(arcsin310)cos(arcsin310)sin(arctanx)cos(arccosy1+y2)+cos(arctanx)sin(arccosy1+y2)cos(arctanx)cos(arccosy1+y2)sin(arctanx)sin(arccosy1+y2)=sin(arcsin310)cos(arcsin310)sin(arctanx)y1+y2+cos(arctanx)sin(arccosy1+y2)cos(arctanx)y1+y2sin(arctanx)sin(arccosy1+y2)=310cos(arcsin310)sin(arctanx)=x1+x2cos(arctanx)=11+x2x1+x2y1+y2+11+x2sin(arccosy1+y2)11+x2y1+y2x1+x2sin(arccosy1+y2)=310cos(arcsin310)sin(arccosz)=1z2cos(arcsinz)=1z2x1+x2y1+y2+11+x21y21+y211+x2y1+y2x1+x21y21+y2=3101910x1+x2y1+y2+11+x211+y211+x2y1+y2x1+x211+y2=310110xy+1yx=3xy+1=3(yx)xy3(yx)+1=0xy+3x3y+1=0a=3b=3c=1

 

laugh

13.07.2021
 #1
avatar+26396 
+2

Find all pairs (x,y) of real numbers such that
x+y=10x2+y2=56+xy.

 

a+b=10b=10a

 

(x+y)2=x2+2xy+y2(x+y)2=x2+y2+2xy|x2+y2=56+xy102=56+xy+2xy102=56+3xy3xy=100563xy=44|:3xy=443|y=10xx(10x)=44310xx2=443x210x+443=0x=10±10244432x=10±42544432x=10±2254432x=5±25443x=5±75443x=5±313

 

1.

x=5+313y=10a=10(5+313)y=5313

 

2.

x=5313y=10a=10(5313)y=5+313

 

laugh

12.07.2021
 #1
avatar+26396 
+2

Find all solutions to the system
a+b=14a3+b3=812+3ab

 

a+b=14b=14a

 

(a+b)3=a3+3a2b+3ab2+b3(a+b)3=a3+b3+3ab(a+b)|a+b=14143=a3+b3+314ab|a3+b3=812+3ab143=812+3ab+314ab143=812+315ab315ab=143812315ab=1932|:315ab=644|b=14a15a(14a)=644a(14a)=6441514aa2=64415a214a+64415=0a=14±1424644152a=14±4494644152a=14±249644152a=7±4964415a=7±73564415a=7±9115

 

1.

a=7+9115b=14a=14(7+9115)b=79115

 

2.

a=79115b=14a=14(79115)b=7+9115

 

laugh

12.07.2021
 #1
avatar+26396 
+3

In the diagram below,

Find PQ.

 

 

 

 

 

 

 

 

laugh

12.07.2021