Find the value of trigonometry

[songoku]

Homework Statement

Find the exact value of cos 247^o+sin 293^o

Homework Equations

trigonometry identities

The Attempt at a Solution

The furthest I can get is:

cos 247^o+sin 293^o
= cos 247^o - cos 23^o
= -2 sin 135^o sin 112^o
= -√2 cos 22^o

Then stuck...

 

[tiny-tim]

hi independent!

cos 247^o+sin 293^o
= cos 247^o - cos 23^o

erm … 203 ?

hint: always subtract from 360° (if necessary), to get an angle less than 180° !

 

[Chestermiller]

247 = 270 - 23

293 = 270 + 23

cos (270 - 23) = ?
sin (270 + 23) = ?

 

[songoku]

hi independent!


erm … 203 ?

hint: always subtract from 360° (if necessary), to get an angle less than 180° !

Hi independent??

Sorry I don't think I get your hint. cos 23^o = - cos 203^o. So
cos 247^o + cos 203^o
= 2 cos 225^ocos 22^o
= -√2 cos 22^o

247 = 270 - 23

293 = 270 + 23

cos (270 - 23) = ?
sin (270 + 23) = ?

cos (270 - 23) = - sin 23^o
sin (270 + 23) = - cos 23^o

So:
- sin 23^o - cos 23^o
= - sin 23^o - sin 67^o
= -2 sin 45^o cos 22^o
= - √2 cos 22^o

All lead to - √2 cos 22^o. Maybe it is me doing it the wrong way but I don't know how to manipulate the question in other way to give a better result

Thanks

 

[tiny-tim]

hi songoku!

(sorry about the typo! )

Hi independent??

Sorry I don't think I get your hint. cos 23^o = - cos 203^o.

ah, i didn't see that

All lead to - √2 cos 22^o. Maybe it is me doing it the wrong way but I don't know how to manipulate the question in other way to give a better result

yes, that's the best you can do

(btw, i'd have done cos247 + sin293 = cos113 - sin67 = -cos67 - sin67 = -√2cos22 …

Chestermiller's way of course is also fine, but I'm not as familiar with cos and sin of 270° as he is, so i'd be afraid of making a mistake! )

i'l guess that when the question said "exact", it just meant "compact"

 

[haruspex]

Homework Statement

Find the exact value of cos 247^o+sin 293^o

Had it said cos 247^o+cos 293^o there would have been an exact answer. (Note what the two angles add up to.) Misprint perhaps?

 

[songoku]

hi songoku!

(sorry about the typo! )


ah, i didn't see that


yes, that's the best you can do

(btw, i'd have done cos247 + sin293 = cos113 - sin67 = -cos67 - sin67 = -√2cos22 …

Chestermiller's way of course is also fine, but I'm not as familiar with cos and sin of 270° as he is, so i'd be afraid of making a mistake! )

i'l guess that when the question said "exact", it just meant "compact"

Had it said cos 247^o+cos 293^o there would have been an exact answer. (Note what the two angles add up to.) Misprint perhaps?

Maybe misprint. Thanks a lot for all the help