## UnkleRhaukus 2 years ago $f(x)=3\sin(x)-2\cos(x)$

1. UnkleRhaukus

\begin{align*} f(x)&=3\sin(x)-2\cos(x)\\ &=\sqrt{(-2)^2+3^2}\cos\Big(x-\text{arctan2}\big({3,-2}\big)\Big)\\ &=\sqrt{13}\cos\Big(x-\big(\pi+\arctan(\tfrac3{-2})\big)\Big)\\ &=\sqrt{13}\cos\left(x-\pi+\arctan(\tfrac3{2})\right)\\ &=\sqrt{13}\sin\big(x-\tfrac\pi2+\arctan(\tfrac3{2})\big)\\ &\approx\sqrt{13}\sin\big(x-0.588\big)\\ \\ &A=\sqrt{13}\\ &\phi=-\frac\pi2+\arctan(\tfrac3{2})\approx -0.588\\ &T=2\pi\qquad\qquad\omega=\tfrac1{2\pi}\\ \end{align*}

2. UnkleRhaukus

3. UnkleRhaukus

right?

4. ZeHanz

It's$f(x)=\sqrt{3^2+(-2)^2}\sin \left( x-\arctan \left( \frac{ 2 }{ 3 } \right) \right)$$\approx \sqrt{13}\sin (x- 0.588)$So you are right, but you made a typo in the equation editor (3/2 instead of 2/3)

5. UnkleRhaukus

i can't see the typo

6. ZeHanz

You wrote$\arctan \left( \frac{ 3 }{ 2 } \right)$It must be:$\arctan \left( \frac{ 2 }{ 3 } \right)$

7. UnkleRhaukus

i have applied these definitions, what method are you using

8. ZeHanz

I applied the same rule. Your answer is right. You just made a typo in your explanation above (in the equation editor). Just carefully look what you typed there (3/2). In your actual calculation you used 2/3, which is right....

9. UnkleRhaukus

i dont think i have made a mistake

10. ZeHanz

If 3/2 was the same as 2/3 you didn't ;)

11. UnkleRhaukus

$-\tfrac\pi2+\arctan(\tfrac32)\approx -0.588$ $\qquad\quad\arctan(\tfrac23)\approx 0.588$

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