Question

Sin(120t)+sin(80t)/cos(120t)-cos(80t)

Answer 1

Ответ:

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A wave travelling along a string is described by,

y(x,t)=0.005sin(80.0x−3.0t), in which the numerical constants are in SI units (0.005 m, 80.0 rad m

−1

and 3.0 rad s

−1

). Calculate :

(a) the amplitude

(b) the wavelength

(c) the period and frequency of the wave.

Also, calculate the displacement y of the wave at a distance x = 30.0 cm and time t = 20 s?

Hard

Solution

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We know that

general equation of wave

y(x,t)=Asin(kx−ωt)

comparing this equation with given equation

1) A=0.005m → amplitude

k=80

ω=3

2) λ=

k

2π

​

=

80

2π

​

=(

40

π

​

)m

3) Time period T=

ω

2π

​

=

3

2π

​

sec

frequency f=

T

1

​

=

2π

3

​

Hz

4) Displacement at x=0.3m

t=20 sec

y=0.005sin(80×0.3−3×20)

=0.005sin(24−60)=0.005sin(−36)

=0.00495m

y=4.9mm.

Объяснение:

Answer 2

To simplify the expression $\frac{\sin(120t) + \sin(80t)}{\cos(120t) - \cos(80t)}$, we can use trigonometric identities. Specifically, we'll use the sum-to-product identities, which state that:

1. $\sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$
2. $\cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

In your expression, $A = 120t$ and $B = 80t$. So, applying the identities:

$$\frac{\sin(120t) + \sin(80t)}{\cos(120t) - \cos(80t)} = \frac{2\sin\left(\frac{120t+80t}{2}\right)\cos\left(\frac{120t-80t}{2}\right)}{-2\sin\left(\frac{120t+80t}{2}\right)\sin\left(\frac{120t-80t}{2}\right)}$$

Now, simplify further:

$$\frac{2\sin(100t)\cos(20t)}{-2\sin(100t)\sin(20t)}$$

The $\sin(100t)$ terms cancel out:

$$\frac{2\cos(20t)}{-2\sin(20t)}$$

Now, you can simplify by canceling a factor of 2:

$$\frac{\cos(20t)}{-\sin(20t)}$$

So, the simplified expression is $-\cot(20t)$, where $\cot(x)$ is the cotangent function.