Question on Inverse Function (1 Viewer)

[lyounamu]

Find the exact values of x and y which satisfy the simultaneous equations
​

sin-1x - cos-1y = pi/12
cos-1x +sin-1x = 5pi/12

I found the answers
x = sqr(3)/2 and y = 1/sqr(2)
but my working out took me quite a long time to work this out.
Because I had to draw four diagrams to make entity (separate entity) for each one.

Can anyone provide me with better working out? (shorter than 1 page if it's okay, mine is 1 page and a half) ​

 

[midifile]

Find the exact values of x and y which satisfy the simultaneous equations
sin-1x - cos-1y = pi/12
cos-1x +sin-1x = 5pi/12

I found the answers
x = sqr(3)/2 and y = 1/sqr(2)
but my working out took me quite a long time to work this out.
Because I had to draw four diagrams to make entity (separate entity) for each one.

Can anyone provide me with better working out? (shorter than 1 page if it's okay, mine is 1 page and a half)

im letting arc = inverse to make it clearer

Adding the two simultaneous equations:
arcsinx + arccosx + arcsiny - arccosy = pi/2
But arcsinx + arccosx = pi/2
therefore pi/2 + arcsiny - arccosy = pi/2
arcsiny - arccosy = 0
arcsiny = arccosy
therefore y = 1/sqrt2

Then sub this into one of the equations and you get x = sqrt3/2

 

[tacogym27101990]

you can use the fact that sin-1x + cos-1x = pi/2

and make substitutions like in sin-1x - cos-1x = pi/12 >>> sin-1x+sin-1y=7pi/12

then do the same for the other equation so u have them both in terms of inverse sines

then add them together

 

[lyounamu]

im letting arc = inverse to make it clearer

Adding the two simultaneous equations:
arcsinx + arccosx + arcsiny - arccosy = pi/2
But arcsinx + arccosx = pi/2
therefore pi/2 + arcsiny - arccosy = pi/2
arcsiny - arccosy = 0
arcsiny = arccosy
therefore y = 1/sqrt2

Then sub this into one of the equations and you get x = sqrt3/2

omg, you are right. Thanks!

 

[vds700]

Find the exact values of x and y which satisfy the simultaneous equations
​

sin-1x - cos-1y = pi/12
cos-1x +sin-1x = 5pi/12

I found the answers
x = sqr(3)/2 and y = 1/sqr(2)
but my working out took me quite a long time to work this out.
Because I had to draw four diagrams to make entity (separate entity) for each one.

Can anyone provide me with better working out? (shorter than 1 page if it's okay, mine is 1 page and a half) ​

i remember doing a question like this in last years 3 unit HSC. Is that where u got it from?

 

[lyounamu]

i remember doing a question like this in last years 3 unit HSC. Is that where u got it from?

Probably not. This is the CSSA paper. I didn't know that

Sin-1(x) + Cos-1(x) = pi/2

This is an awesome knowledge.

 

[midifile]

Sin-1(x) + Cos-1(x) = pi/2

This is an awesome knowledge.

Even if you forget it, you can always work it out using a right angled triangle