Trigonometric Identities

**jessm001** · April 7th 2012, 08:37 AM

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

**Prove It** · April 7th 2012, 08:45 AM

Originally Posted by jessm001

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

Use the Pythagorean Identity to get the values for cos(x) and sin(y), then apply the angle sum identity you attempted to use

**a tutor** · April 7th 2012, 08:46 AM

From $\cos^2 x + \sin ^2 x =1$ we get $\cos x = \pm \sqrt{1-\sin^2 x}$ .

**jessm001** · April 7th 2012, 08:57 AM

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 ?

**Prove It** · April 7th 2012, 06:09 PM

Originally Posted by jessm001

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 ?

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

**Soroban** · April 8th 2012, 08:04 AM

Hello, jessm001!

You were on your way.
We are expected to find those missing values.

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

$\sin x \:=\:\frac{2}{3} \:=\:\frac{opp}{hyp} \qquad x\text{ is in Quadrant 1 or 2.}$
. . $adj \:=\:\pm\sqrt{5} \quad\Rightarrow\quad \cos x \:=\:\pm\frac{\sqrt{5}}{3}$

$\cos y \:=\:\frac{\text{-}2}{7} \:=\:\frac{adj}{hyp}\qquad y\text{ is in Quadrant 2 or 3.}$
. . $opp \:=\:\pm\sqrt{45} \:=\:\pm3\sqrt{5} \quad\Rightarrow\quad \sin y \:=\:\pm\frac{3\sqrt{5}}{7}$

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

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

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

**jessm001** · April 8th 2012, 09:09 AM

Thanks a lot !

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