Question

WILL FAN AND MEDAL
Find the exact value of cos(α+β) under the given conditions. sinα=15/17, 0<α<π/2​; cosβ=5/13, 0<β<π/2

Answer 1

you have to use \[\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)\]

Answer 2

two of those numbers you are told
you have to find the other two

Answer 3

do you know how to do it?

Answer 4

So, if I plug in the numbers, the equetion would be\[\cos( \alpha+5)=\cos(\alpha)\cos(5)-\sin(15/17)\sin\]

Answer 5

oh no

Answer 6

I forgot to add the beta but I think you get what I'm saying, right?

Answer 7

lets go slow

Answer 8

it is not the sine of 15/17, the sine IS 15/17

Answer 9

lets see which two numbers you know, and which two you need

Answer 10

Oh, okay.

Answer 11

5 and 15/17 are the two numbers I have

Answer 12

because of \[\sin \alpha^2 + \cos \alpha^2 = 1\]

Answer 13

you can know other two by this equation

Answer 14

\[\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)\]\[\cos(\alpha+\beta)=\cos(\alpha)\times\frac{5}{13}-\frac{15}{17}\times \sin(\beta)\]

Answer 15

i substituted directly in the formula the two numbers you were given

Answer 16

Oh okay. Thanks

Answer 17

what is missing? two more numbers, namely \[\cos(\alpha)\] and \[\sin(\beta)\]

Answer 18

And how exactly do I find those?

Answer 19

you are told that \[\sin(\alpha)=\frac{15}{17}\] and you need \(\cos(\alpha)\)
do you know how to find it?

Answer 20

there are a couple ways
method one
use \[\cos^2(\alpha)+\sin^2(\alpha)=1\]
i find it simpler to draw a triangle

Answer 21

Well, sin is Opposite over Hyp and Cosine is Adj. over hypotenuse so I know that the bottom number of the fraction would be 17

Answer 22

solve for \(a\) using pythagoras , or from memory

Answer 23

Okay ignore what I said

Answer 24

yeah what you said is right

Answer 25

I see what you're saying

Answer 26

just need the missing side

Answer 27

Oh it is? Cool

Answer 28

let me know what you get

Answer 29

I got 8

Answer 30

Is that right or am I epically failing right now?

Answer 31

yes so\[\cos(\alpha)=\frac{8}{15}\]

Answer 32

\[\cos(\alpha+\beta)=\frac{8}{17}\times\frac{5}{13}-\frac{15}{17}\times \sin(\beta)\]

Answer 33

Got it. So now I need the exact value of cos(alpha+beta)

Answer 34

Okay you're ahead of me

Answer 35

ok i made a typo, should be \[\cos(\alpha)=\frac{8}{\color{red}{17}}\]

Answer 36

Alright. And then I cross multiply, right?

Answer 37

now find \[\sin(\beta)\] in a similar way

Answer 38

there is no such thing as "cross multiply"
you need one more number

Answer 39

Oh yeah I just saw that. Sorry about that I'm so lost on this

Answer 40

it is just right triangle is all
pythagoras all the way

Answer 41

So, since sin alpha is 15/17 I need to draw another triangle with sides that equal 15 and 17, right?

Answer 42

Wait

Answer 43

we got that one already

Answer 44

\[\cos(\alpha+\beta)=\frac{8}{17}\times\frac{5}{13}-\frac{15}{17}\times \sin(\beta)\]what number is missing?

Answer 45

So we do the one with the sides of 5 and 13

Answer 46

sin beta

Answer 47

yes

Answer 48

Okay I'll draw that

Answer 49

Would it look like that?

Answer 50

yes

Answer 51

And the missing side is 12

Answer 52

yes

Answer 53

\[\cos(\alpha+\beta)=\frac{8}{17}\times\frac{5}{13}-\frac{15}{17}\times \frac{12}{13}\]

Answer 54

So the fraction would be 12/13

Answer 55

yes

Answer 56

Oh you're one step ahead of me

Answer 57

yes

Answer 58

And then do I simplify the equation or....?