Question

Please help! I have been trying to work this out for an hour and I am going crazy. I will award and medal!!!
If sinxcos(pi/7)-sin(pi/7)cosx=-(sqrt2)/2, then x can equal __; check all that apply

A. 3pi/4+pi/7+2npi
B. 7pi/4+pi/7+2npi
C. pi/4+pi/7+2npi
D. 5pi/4+pi/7+2npi

Answer 1

look at this list of trig identities

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

look on page 2 at the section `Sum and Difference Formulas`

Answer 2

which of those identities in that section will we use?

Answer 3

any ideas @lina777 ?

Answer 4

Thank you for answering! @jim_thompson5910 The identity we will use is sin(a-b)=sinacosb+cosasinb

Answer 5

close

Answer 6

it's actually

sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B)

you will have a minus on the right side. Not a plus

Answer 7

that means `sin(x)*cos(pi/7) - cos(x)*sin(pi/7)` will turn into `sin(x-pi/7)`

Answer 8

so
sin(x-pi/7) = -sqrt(2)/2
are you able to see what comes next?

Answer 9

hint: it will involve the unit circle

Answer 10

Hello @jim_thompson5910 I got the stomach flu, I am sorry for not responding. Can you still help me out?

Answer 11

so we left off at `sin(x-pi/7) = -sqrt(2)/2` right?

Answer 12

Right. :)

Answer 13

Now use the unit circle
https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/2000px-Unit_circle_angles_color.svg.png
to figure out which points have a y coordinate of \(\Large \frac{-\sqrt{2}}{2}\)

Hint: there are 2 of these points. One is in Q3 and the other is in Q4

Answer 14

7pi/4 and 5pi/4

Answer 15

let's just focus on the 7pi/4 for now

Answer 16

So is that the answer? 7pi/4+pi/7+2npi and 5pi/4+pi/7+2npi

Answer 17

sin(x-pi/7) = -sqrt(2)/2

will mean that

x-pi/7 = 7pi/4+2n*pi

the `+2n*pi` part ensures that we capture all of the coterminal angles. The n represents any integer (whole number).

Answer 18

`So is that the answer? 7pi/4+pi/7+2npi and 5pi/4+pi/7+2npi `
yes

Answer 19

Wow, that was much simpler than I thought it was! Thank you very much. And that is for sure?

Answer 20

`And that is for sure? `

yes, I used a graphing calculator to confirm

Answer 21

Thank you so much for your help!!

Answer 22

no problem