# trigonometry question

• Oct 3rd 2009, 06:17 AM
thereddevils
trigonometry question
solve for x , -180<x<180

2cos x+1= sin x

This is what i did ..

sin x -2 cos x=1

$\displaystyle \sqrt{5}\sin (x-63.43)=1$

so x-63.43=26.57

x=90

and x-63.43=-26.57 , -153.43

x=36.86 , -90

My answers are in red . But tat doesn't agree with the answer in the book . Thanks for helping .
• Oct 3rd 2009, 10:33 AM
Jameson
Quote:

Originally Posted by thereddevils
sin x -2 cos x=1

$\displaystyle \sqrt{5}\sin (x-63.43)=1$

so x-63.43=26.57

How did you go from the first line to the second?
• Oct 3rd 2009, 10:34 AM
apcalculus
Quote:

Originally Posted by thereddevils
solve for x , -180<x<180

2cos x+1= sin x

This is what i did ..

sin x -2 cos x=1

$\displaystyle \sqrt{5}\sin (x-63.43)=1$

so x-63.43=26.57

x=90

and x-63.43=-26.57 , -153.43

x=36.86 , -90

My answers are in red . But tat doesn't agree with the answer in the book . Thanks for helping .

2 cos x + 1 = sin x
$\displaystyle 2 \cos x + 1 = \sqrt{1 - \cos^2x}$

Squaring both sides, you get:

$\displaystyle 4 \cos^2x+ 4 \cos x = 1 - \cos^2 x$

Move everything to one side:
$\displaystyle 5\cos^2x + 4 \cos x - 1 = 0$
Sub: $\displaystyle u = \cos x$
$\displaystyle 5u^2 + 4u -1 = 0$

$\displaystyle u = \frac{-4 \pm 6}{10}$

giving -1 or 0.2.

Set cosine x equal to each and solve. Check your solutions to make sure they work.

Good luck!!
• Oct 3rd 2009, 09:36 PM
thereddevils
Quote:

Originally Posted by Jameson
How did you go from the first line to the second?

I used this r sin (x-a) , where $\displaystyle r=\sqrt{1^2+2^2}=\sqrt{5}$

and tan a = 2

do you see anything wrong in my working ?
• Oct 4th 2009, 01:20 AM
mr fantastic
Quote:

Originally Posted by thereddevils
solve for x , -180<x<180

2cos x+1= sin x

This is what i did ..

sin x -2 cos x=1

$\displaystyle \sqrt{5}\sin (x-63.43)=1$

so x-63.43=26.57

x=90 Mr F says: This is correct.

and x-63.43=-26.57 , -153.43 Mr F says: This is wrong. It should be x - 63.43 = 180 - 26.57 = 153.43. Note that sin(-A) = -sin(A) .....

[snip]

Therefore x - 63.43 = 153.43 => x = 216.86 degrees which is outside the domain therefore subtract 360 degrees to get the other value of x.