Question

Use the given information to evaluate cos(α+β).

Answer 1

is there any extra info about the angle \(\large \beta\)?

Answer 2

No that's all that was in the question

Answer 3

-8/15 doesn't land it on the IV quadrant
it lands it on either the II or III one

and depending on that..... its sine could be either positive or negative
so that's a bit ambiguous

Answer 4

the angle \(\large \alpha\) has a sine of -7/25 is negative
the onlyl two possibilities are III and IV quadrants

it says is not on the IV one, so is in the III
so that's easy

but the angle \(\large \beta\) is rather ambiguous though

Answer 5

no i dont agree that b is ambiguous

Answer 6

b has to be in quadrant 2

Answer 7

hmmm why not quadrant III?

Answer 8

in quadrant 3, tangent is positive, and thus cotangent must also be positive

Answer 9

from this, you can also deduce that \[\tan b=\frac{-15}{8}\]

Answer 10

We also know that \[\tan x=\frac{\sin x}{\cos x}\]

Answer 11

but first, \[\cos (a+b)=\cos a \cos b - \sin a \sin b\]

Answer 12

do you know how to find cos (a)?

Answer 13

hmmm hmm? we don't have info on tangent though
we only know the adjacent side is -8 and they hypotenuse is 25

the hypotenuse could swivel to either quadrant, giving a cosine of -8/25 anyway

Answer 14

I'm not sure how to find cos(a)

Answer 15

im working on that tangent too, its kinda difficult......

Answer 16

ohhhhhhhh i got it

Answer 17

ok, so I'm going to start with angle b because its harder

Answer 18

okay

Answer 19

are you familiar with the identity \[1+\cot^2 x=\csc^2 x\]

Answer 20

a little bit

Answer 21

can you square that "cot b" for me

Answer 22

square (-8/15)

Answer 23

64/225?

Answer 24

yes, now add 1 to that, to get csc^2 b

Answer 25

289/225?

Answer 26

yes, now take the square root of that to get csc b

Answer 27

17/15?

Answer 28

yes. knowing that \[\csc b = \frac{1}{\sin b}\] find Sin b

Answer 29

okay so i know to work backwards. do i just multiply 1(17/15)?

Answer 30

you need to take the reciprical of 17/15

Answer 31

15/17?

Answer 32

yes

Answer 33

now, you know that sin b = 15/17. Now, plug this into the identity \[\cos^2 (b) + \sin^2 (b) = 1\]

Answer 34

and solve for cos b

Answer 35

first, whats sin^2 b

Answer 36

um sin^2b is sin^2(15/17), right? so it's .5963?

Answer 37

no that is not what i wanted. sin b = 15/17 so sin^2 b = (15/17)^2. solve (15/17)^2

Answer 38

225/289?

Answer 39

yes now subtract that from 1

Answer 40

64/289

Answer 41

take the square root of that to get cos b

Answer 42

8/17

Answer 43

You need the negative root because angle b is in the 2nd quadrant, where cosine is negative

Answer 44

so once you find that, write cos b and sin b out here:

Answer 45

how do i find the negative root?

Answer 46

\[\cos^2 b = \frac{64}{289}\] SO\[\cos b = \pm \sqrt{\frac{64}{225}}=\pm \frac{8}{17}\] IN THIS CASE WE NEED NEGATIVE ROOT:\[\large \cos b = \frac{-8}{17}\]

Answer 47

now write what i told you

Answer 48

sinb=15/17 and cosb-8/17

Answer 49

ok now onto angle a. We know a is in quadrant 3, where sin and cos are negative. Just like before, i want you to try and use the identity to get cos a, using the sin a provided

Answer 50

okay first is -7/25^2= 49/625

Answer 51

then (49/625)+1= 674/625

Answer 52

no, u need to subtract 49/625 from 1

Answer 53

oh yeah sorry. 1-(49/625)= -(576/625)

Answer 54

its not negative , its just positive 576/625. now take the square root of that

Answer 55

square root of (576/625) is 24/25

Answer 56

you need the negative root because cos is negative in quadrant III. now list cos a and sin a

Answer 57

sin a=-7/25 and cos a= -24/25

Answer 58

cool, relist sin b and cos b for convenience

Answer 59

sin b=15/17 and cos b=-8/17

Answer 60

\[\cos (a+b) = \cos a \cos b -\sin a \sin b\] PLug in values and solve

Answer 61

cos(-24/25+-8/17)=((-24/25)(-8/17)-(-7/25)(15/17))

Answer 62

cos(-24/25+-8/17)= 297/425?

Answer 63

i didnt get that. Plug this into calculator : \[\huge \frac{24\times8+7\times15}{17\times25}\]

Answer 64

20,160/425?

Answer 65

ok, thats a plus sign, not a multiplication sign

Answer 66

ohhh I'm sorry. 297/425? that's what i got before

Answer 67

yeah sorry you were right. Medal plz! Fan PLz

Answer 68

okay thank you so much! :)

Answer 69

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