1. ## Trig

Hey.

1- Express 12cosx+ 9sinx in the form Rcos(x-A) where R is greater than 0 and A is between 0 and 90.

I did this one and got 15cos(x-0,644) (in radians)

I cant do the second part.

b) Use the method of part a to find the smallest positve root of A of the equation 12cosx+9sinx = 14

Thanks.

2. Hello Oasis1993
Originally Posted by Oasis1993
Hey.

1- Express 12cosx+ 9sinx in the form Rcos(x-A) where R is greater than 0 and A is between 0 and 90.

I did this one and got 15cos(x-0,644) (in radians)

I cant do the second part.

b) Use the method of part a to find the smallest positve root of A of the equation 12cosx+9sinx = 14

Thanks.
You're right so far!

For part (b), just say:
$\displaystyle 15\cos(x-A) = 14$

$\displaystyle \Rightarrow \cos(x-A)=\frac{14}{15}= 0.9333$

$\displaystyle \Rightarrow x-A = \pm 0.3672 +2n\pi, n \in \mathbb{Z}$ (Do you understand this bit?)

$\displaystyle \Rightarrow x = A \pm 0.3672 + 2n\pi$
And for the smallest positive value of $\displaystyle x$, which sign do we take, and which value of $\displaystyle n$? (I get the answer $\displaystyle x=0.277$. Do you?)

3. Thank you!

The principal value of $\displaystyle (x - A)$ is $\displaystyle 0.3672$ radians. Other angles that will have the same cosine as this will be $\displaystyle 0.3672$ radians on either side of a multiple of $\displaystyle 2\pi$. Any multiple of $\displaystyle 2\pi$ is $\displaystyle 2n\pi$, and 'on either side' of this means adding or subtracting $\displaystyle 0.3672$ from $\displaystyle 2n\pi$. Hence $\displaystyle \pm0.3672 + 2n\pi$.