Question

find cos (x + y) if sin x = 5/13 and sin y = 4/5

Answer 1

do you know the cosine sum of angles formula?

Answer 2

I'm afraid I don't.

Answer 3

cos(x + y) = cos x cos y - sin x sin y

Answer 4

I don't know cosine of either of these though.

Answer 5

ah, yes you do, you just don't realize it :-)

Remember, sin^2 x + cos^2 x = 1

so if you know sin x, you can find cos x by rearranging that equation and substituting in the value you know for sin x

Answer 6

cos^2 x = 1 - sin ^2 x

Answer 7

if sin x = 5/13, then

sin^2 x + cos^2 x = 1

(5/13)^2 + cos^2 x = 1
cos^2 x = 1-(5/13)^2
cos x = sqrt(1-(5/13)^2)

Answer 8

and if you actually expand the stuff inside the square root, you'll get something for which the square root can be easily found

Answer 9

I'm just too lazy to type the steps, sorry :-)

Answer 10

oh, what the heck...

sqrt of (1 - (25/169)) = sqrt (169/169 - 25/169) = sqrt (144/169) = 12/13

Answer 11

12/13?

Answer 12

good timing, haha

Answer 13

yep!

now do the same for the other one

Answer 14

cos x^2 = sqrt 1 - sin x^2
= sqrt 1 - 4/5
= sqrt 25/25 - 16/25
= 9/25

Answer 15

sqrt (9/25) = 3/5

Answer 16

when you take the square root, should consider both solutions. I.e plus or minus?

Answer 17

ah, didn't square root the last part. now how do i find cosine of x and y? do i just add 12/13 and 3/5?

Answer 18

now you have all the pieces to use the identity I gave you

Answer 19

cos (x+y) = cos x * cos y - sin x * sin y

cos x = 12/13
cos y = 3/5
sin x = 5/13
sin y = 4/5

Answer 20

@sourwing not necessary to consider the negative square root — these the side lengths of the triangle in the unit circle

Answer 21

i got .246 when i plugged that into my calculator

Answer 22

that's about right, the exact answer is 16/65

Answer 23

should definitely do the exact answer in a problem like this

Answer 24

alrighty thank you so much! i have two more with different identities that i don't know, if you wouldn't mind helping.

Answer 25

@whpalmer4 can you help me on my question

Answer 26

well, the sides length are positive, but their coordinate is not necessarily positive

Answer 27

sin (x - y) if cos(x) = 8/17 and cos(y) = 3/5

tan(x - y) if csc(x) = 13/5 and cot(y) = 4/3

Answer 28

okay, here are the identities:

sin(x-y) = sin x cos y - cos x sin y

tan x - y = (tan x - tan y)/(1 + tan x tan y)

csc x = 1/sin x
cot x = 1/tan x = cos x / sin x

Answer 29

a handy listing of trig identities that I refer to often can be found at http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Answer 30

ok, so if sin x ^2 + cos x^2 = 1, and i'm given cos, then
sin x^2 = sqrt 1-cos(x)^2, right?

Answer 31

close, sin x = sqrt (1 - cos^2 (x))

Answer 32

square root 289/289 - 64/289

square root 225/289

15/17

Answer 33

square root 1 - 9/25
square root 25/25 - 9/25
square root 16/25
4/5

Answer 34

yes, those are both correct

Answer 35

12/25 - 40/63?

Answer 36

oops wait messed that one up

Answer 37

12/25 - 120/289, can't figure out how to simplify 120/289

Answer 38

i guess i cant lol since the only factors of 289 are 1, 17, and 289

Answer 39

cos x = 8/17
cos y = 3/5
sin x = 15/17
sin y = 4/5

sin(x-y) = sin x cos y - cos x sin y
= 15/17 * 3/5 - 8/17 * 4/5
= (45 - 32) / (5*17) = 13/85

Answer 40

lol thank you

Answer 41

how about that tan question? I'm fading quickly, act now before I fall asleep :-)

Answer 42

I have done everything and got to the last step and see the answer is 16/65 which is .246 but i get .2288 somehow.