# Thread: reduction of a cosΘ+/_ b sinΘ

1. ## reduction of a cosΘ+/_ b sinΘ

reduct this problem..

1/2 sinθ + /2 cos Θ

to the form

a. r sin (Θ+Φ)
b. r cos(Θ+Φ)

2. Originally Posted by jasonlewiz
reduct this problem..

1/2 sinθ + /2 cos Θ

to the form

a. r sin (Θ+Φ)
b. r cos(Θ+Φ)
$\displaystyle \cos\left(\frac{\pi}{3}\right) \sin(\theta) + \sin\left(\frac{\pi}{3}\right) \cos(\theta)$

or

$\displaystyle \sin\left(\frac{\pi}{6}\right) \sin(\theta) + \cos\left(\frac{\pi}{6}\right) \cos(\theta)$

finish up by using the sum/difference identities for sine and cosine

3. ## Handling expressions of the form a sin x + b cos x

Hello jasonlewiz
Originally Posted by jasonlewiz
reduct this problem..

1/2 sinθ + /2 cos Θ

to the form

a. r sin (Θ+Φ)
b. r cos(Θ+Φ)
skeeter's solution is all you need for this particular problem. But, in case you don't spot that $\displaystyle \tfrac12$ can be written as $\displaystyle \cos\tfrac{\pi}{3}$ and that $\displaystyle \tfrac{\sqrt3}{2}$ is the sine of the same angle ($\displaystyle \tfrac{\pi}{3}$), here's the more general method for handling this type of expression:

$\displaystyle \tfrac12\sin\theta + \tfrac{\sqrt3}{2}\cos\theta = r \sin(\theta + \phi)$

$\displaystyle = r\sin\theta\cos\phi + r\cos\theta\sin\phi$

Compare coefficients of $\displaystyle \sin\theta$ and $\displaystyle \cos\theta$:

$\displaystyle \tfrac12 = r \cos\phi$ (1)

$\displaystyle \tfrac{\sqrt3}{2} = r\sin\phi$ (2)

Divide (2) by (1): $\displaystyle \tan\phi = \sqrt3$

$\displaystyle \Rightarrow \phi = \tfrac{\pi}{3}$

Square (1) and (2) and add:

$\displaystyle \tfrac14 + \tfrac34 = r^2(\sin^2\phi + \cos^2\phi)=r^2$

$\displaystyle \Rightarrow r = 1$ (taking the positive root)

So there are the values of $\displaystyle r$ and $\displaystyle \phi$.

You can do part (b) in exactly the same way, but starting with $\displaystyle \tfrac12\sin\theta + \tfrac{\sqrt3}{2}\cos\theta = r \cos(\theta + \phi)$. (In fact, it's slightly easier to start with $\displaystyle \tfrac12\sin\theta + \tfrac{\sqrt3}{2}\cos\theta = r \cos(\theta - \phi)$ since $\displaystyle \phi$ then turns out to be a positive angle.)