If sin x = 2/3 and cos y = -2/7, find the possible values of cos(x+y). Anyone please?

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

= -2/7cos x - 2/3 sin y

now i'm stuck

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- Apr 7th 2012, 08:37 AMjessm001Trigonometric Identities
If sin x = 2/3 and cos y = -2/7, find the possible values of cos(x+y). Anyone please?

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

= -2/7cos x - 2/3 sin y

now i'm stuck - Apr 7th 2012, 08:45 AMProve ItRe: Trigonometric Identities
- Apr 7th 2012, 08:46 AMa tutorRe: Trigonometric Identities
From $\displaystyle \cos^2 x + \sin ^2 x =1 $ we get $\displaystyle \cos x = \pm \sqrt{1-\sin^2 x}$.

- Apr 7th 2012, 08:57 AMjessm001Re: Trigonometric Identities
I used the pythagorean identity and found values for cos x and sin y, should I now substitute both negative and positive values in the sum angle identity? And should the compound angle identity be equal to something ?

- Apr 7th 2012, 06:09 PMProve ItRe: Trigonometric Identities
Since you know that sin(x) is positive, that means x could be in the first and second quadrants, which means cos(x) could be positive or negative.

Since you know that cos(y) is negative, that means y could be in the second or third quadrants, which means sin(x) could be positive or negative.

So yes, since there are four possibilities, you will need to substitute all four values to get four possible results :) - Apr 8th 2012, 08:04 AMSorobanRe: Trigonometric Identities
Hello, jessm001!

You were on your way.

We are expected to find those missing values.

Quote:

$\displaystyle \text{If }\sin x = \tfrac{2}{3}\text{ and }\cos y = \text{-}\tfrac{2}{7}\text{, find the possible values of }\cos(x+y). $

$\displaystyle \sin x \:=\:\frac{2}{3} \:=\:\frac{opp}{hyp} \qquad x\text{ is in Quadrant 1 or 2.}$

. . $\displaystyle adj \:=\:\pm\sqrt{5} \quad\Rightarrow\quad \cos x \:=\:\pm\frac{\sqrt{5}}{3}$

$\displaystyle \cos y \:=\:\frac{\text{-}2}{7} \:=\:\frac{adj}{hyp}\qquad y\text{ is in Quadrant 2 or 3.}$

. . $\displaystyle opp \:=\:\pm\sqrt{45} \:=\:\pm3\sqrt{5} \quad\Rightarrow\quad \sin y \:=\:\pm\frac{3\sqrt{5}}{7}$

$\displaystyle \cos(x + y) \;=\;\cos x\cos y - \sin x\sin y$

. . . . . . . . $\displaystyle =\;\left(\pm\frac{\sqrt{5}}{3}\right)\left(-\frac{2}{7}\right) - \left(\frac{2}{3}\right)\left(\pm\frac{3\sqrt{5}}{ 7}\right)$

There are four possible values: . $\displaystyle \frac{8\sqrt{5}}{21},\;-\frac{8\sqrt{5}}{21},\;\frac{4\sqrt{5}}{21},\;-\frac{4\sqrt{5}}{21}$

- Apr 8th 2012, 09:09 AMjessm001Re: Trigonometric Identities
Thanks a lot ! :)