# Thread: Finding the amplitude

1. ## Finding the amplitude

I have

$
y=5 sin (4t) + 9 Cos (4t)
$

i am told to put it in the form; $R Sin (4t + \alpha)$where $0 < \alpha < (\pi/2)$

angular freq, w= 4 rads,

and the period is $t= 2 \pi/w$ = 1.57 secs

how to you find the amplitude???

2. Originally Posted by madmax29
I have

$
y=5 sin (4t) + 9 Cos (4t)
$

i am told to put it in the form; $R Sin (4t + \alpha)$where $0 < \alpha < (\pi/2)$

angular freq, w= 4 rads,

and the period is $t= 2 \pi/w$ = 1.57 secs

how to you find the amplitude???
Trigonometry!

Put $\alpha=\arctan(9/5)$ then:

$y=\sqrt{5^2+9^2} \cos(\alpha) \sin (4t) + \sqrt{5^2+9^2} \sin(\alpha) \cos (4t)$

etc..

CB

3. Originally Posted by CaptainBlack
Trigonometry!

Put $\alpha=\arctan(9/5)$ then:

$y=\sqrt{5^2+9^2} \cos(\alpha) \sin (4t) + \sqrt{5^2+9^2} \sin(\alpha) \cos (4t)$

etc..

CB
Therefore; $\alpha=\arctan(9/5)= \arctan(1.8)= 60.94$ in degrees, then:

Convert to radians; $60.94 (\pi/180)=1.06$

Now;

$y=\sqrt{5^2+9^2} \cos(\alpha) \sin (4t) + \sqrt{5^2+9^2} \sin(\alpha) \cos (4t)$

$y=\sqrt{106} \cos(1.06) \sin (4(1.57)) + \sqrt{106} \sin(1.06) \cos (4(1.57))$

$y=10.29 .(0.489) \sin (6.28) + 10.29 .(0.873) \cos (6.28)$

$y=10.29 . (0.489) . (-0.003) + 10.29 . (0.873) . (0.99)$

$y=-0.0151 + 8.893$

$y=8.88$.....

is this correct?, would y be the answer for the amplitude?

4. Found this;

where y from the question = Ao on the diagram above

x-axis would be time (in secs)

5. Originally Posted by madmax29
Therefore; $\alpha=\arctan(9/5)= \arctan(1.8)= 60.94$ in degrees, then:

Convert to radians; $60.94 (\pi/180)=1.06$

Now;

$y=\sqrt{5^2+9^2} \cos(\alpha) \sin (4t) + \sqrt{5^2+9^2} \sin(\alpha) \cos (4t)$

$y=\sqrt{5^2+9^2} \sin (4t+\alpha)$

I don't know what you think the rest means:-

$y=\sqrt{106} \cos(1.06) \sin (4(1.57)) + \sqrt{106} \sin(1.06) \cos (4(1.57))$

$y=10.29 .(0.489) \sin (6.28) + 10.29 .(0.873) \cos (6.28)$

$y=10.29 . (0.489) . (-0.003) + 10.29 . (0.873) . (0.99)$

$y=-0.0151 + 8.893$

$y=8.88$.....

is this correct?, would y be the answer for the amplitude?
No the amplitude is $\sqrt{5^2+9^2} \approx 10.3$

CB

6. Originally Posted by CaptainBlack
$y=\sqrt{5^2+9^2} \sin (4t+\alpha)$

I don't know what you think the rest means:-

No the amplitude is $\sqrt{5^2+9^2} \approx 10.3$

CB
I accept that the amplitude is $R=\sqrt{5^2+9^2} \approx 10.3$

because in the form $y=R Sin (4t + \alpha)=\sqrt{5^2+9^2} \sin (4t+\alpha)$
this in a sense is $y=Asin(t)$, where $A$ is the amplitude

but what does "y" mean in the question, what i have evaluated to make 8.88?

7. Originally Posted by madmax29
I accept that the amplitude is $R=\sqrt{5^2+9^2} \approx 10.3$

because in the form $y=R Sin (4t + \alpha)=\sqrt{5^2+9^2} \sin (4t+\alpha)$
this in a sense is $y=Asin(t)$, where $A$ is the amplitude

but what does "y" mean in the question, what i have evaluated to make 8.88?
You are confusing the variable $t$ which usually represents time with the constant $\tau$ which represents the period (which has the same units at $t$)

CB