# find sine and cosine function from graph

I've been asked to write two equation(one sine and one cosine) for the following graph

I'm understand axis of symmetry is $y=-10$, Period is$\frac{\pi}{30}$ and amplitude is $6$, are these values correct ? how will get and equation of sine and cosine from the graph.

Any help is appreciated, also any resource to learn this topic further will be helpful,

Thank you, Arif

Wave-form in standard form

$$(y-k)= A \sin \dfrac{ 2 \pi (x-h)}{\lambda} = A \cos{ [\frac{\pi}{2} -\dfrac{ 2 \pi (x-h)}{\lambda}]}$$

where we get $(h,k)$ as average values of sine wave inflection point ( below where you marked $15$) with maximum positive slope using the given crest and trough of the sine-wave for $(x-,y-)$ coordinates to determine shifts/translations of a rigid sine curve.

$$k=\frac{-4-16}{2} = -10,\, A=6, \, h= \frac{6-24}{2} = -9, \lambda=60 \,$$

• please can you elaborate on as how we get the values of h and k. – Arif Feb 24 '18 at 14:13
• Apologies, an error occured by oversight. Corrected it, explained in the answer. – Narasimham Feb 24 '18 at 14:56

Period is $30$ and amplitude $6$ thus the function is in the form

$$f(x)=6 \cos \left(\frac{2\pi}{30}x+\phi\right)+C$$

where

• C<10 is the constant for the vertical shifting
• $\phi\approx -\frac{\pi}6$ is the constant for the horizontal shifting

I tried using $Bx+C=0$, our sine function starts at $-24$ and $B=\frac{\pi}{30}$ substituting $x=-24, ~B=\frac{\pi}{30}$ in $Bx+C=0$ we get $C=\frac{24\pi}{30}$

our equation become $6\sin(\frac{\pi}{30}x+\frac{24\pi}{30})-10$. Is this correct ?

Thanks, Arif