Thread: [SOLVED] Trig. and Sin Wave Question

If anyone can help me with these two questions I would appreciate it. Thank you!

Trig. question:
"The expression $\displaystyle [2cos(theta)/sin2(theta)]$ is equivalent to...?
Choices:
(1) [csc theta]
(2) [sec theta]
(3) [cot theta]
(4) [sin theta]

Work done so far:
$\displaystyle [Cos2 theta/Sin2 theta] = [1-sin^2 theta/1-cos^2 theta] $
...where to go from here?

Sin Wave Question:
"A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation $\displaystyle y = a sin bx + c$. The range of the rope’s height above the ground is between 1 and 5 feet. The period of the wave is $\displaystyle [4(pie symbol)]$. Write the equation that represents this wave."


Work done so far:
Amp.= 3
Freq.= 2
Range=...5?

Equation: 3 sin 2x + ...?

Side Note: I'm new to using "LaTex"...

Originally Posted by MrOats

If anyone can help me with these two questions I would appreciate it. Thank you!

Trig. question:
"The expression $\displaystyle [2cos(theta)/sin2(theta)]$ is equivalent to...?
Choices:
(1) [csc theta]
(2) [sec theta]
(3) [cot theta]
(4) [sin theta]

Work done so far:
$\displaystyle [Cos2 theta/Sin2 theta] = [1-sin^2 theta/1-cos^2 theta] $
...where to go from here?

Sin Wave Question:
"A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation $\displaystyle y = a sin bx + c$. The range of the rope’s height above the ground is between 1 and 5 feet. The period of the wave is $\displaystyle [4(pie symbol)]$. Write the equation that represents this wave."


Work done so far:
Amp.= 3
Freq.= 2
Range=...5?

Equation: 3 sin 2x + ...?

Side Note: I'm new to using "LaTex"...

For the first one

$\displaystyle \sin(2\theta)=2\sin(\theta)\cos(\theta)$

$\displaystyle \therefore\frac{2\cos(\theta)}{\sin(2\theta)}=\fra c{2\cos(\theta)}{2\sin(\theta)\cos(\theta)}$

I think you can take it from there

Originally Posted by MrOats

If anyone can help me with these two questions I would appreciate it. Thank you!

Trig. question:
"The expression $\displaystyle [2cos(theta)/sin2(theta)]$ is equivalent to...?
Choices:
(1) [csc theta]
(2) [sec theta]
(3) [cot theta]
(4) [sin theta]

Work done so far:
$\displaystyle [Cos2 theta/Sin2 theta] = [1-sin^2 theta/1-cos^2 theta] $
...where to go from here?

Sin Wave Question:
"A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation $\displaystyle y = a sin bx + c$. The range of the rope’s height above the ground is between 1 and 5 feet. The period of the wave is $\displaystyle [4(pie symbol)]$. Write the equation that represents this wave."


Work done so far:
Amp.= 3
Freq.= 2
Range=...5?

Equation: 3 sin 2x + ...?

Side Note: I'm new to using "LaTex"...

It moves from 1 to 5 for the second one...so amplitude is 4

Since in the equation $\displaystyle a\sin(bx+c)$

The period is $\displaystyle \frac{2\pi}{|b|}$

So you have $\displaystyle \frac{2\pi}{|b|}=4\pi$

Solve for b and consider that the lowest point is one and the highest point is 5 to figure out the c value which just shifts the sine wave vertically

Please check, someone, to make sure I did these right? Thank you!

For this question:
I got $\displaystyle {cos(theta)/sin(theta)}$, which is $\displaystyle cot(theta)$.

Am I correct? If not, please help me.

For the second question:
I got Amplitude = 4, Period = [2(pie)/b], Frequency = 2/4
...Not so sure about this one...

And for the equation: y= 4 Sin (2/4(x)+4)

...I really am unsure about this one. Someone please clarify for me? I need to get "solid" in my head what the amplitude, frequency, period, and range are...And solid, a=, b=, c=...answers.

Please help me demystify this sin wave question! Thank you.

Originally Posted by MrOats

Please check, someone, to make sure I did these right? Thank you!

For this question:
I got $\displaystyle {cos(theta)/sin(theta)}$, which is $\displaystyle cot(theta)$.

Am I correct? If not, please help me.

No you are wrong...

$\displaystyle \frac{2\cos \theta}{\sin 2\theta} = \frac{2\cos \theta}{2\sin \theta\cos \theta} = \frac1{\sin\theta} = csc\theta$

So answer is option (1)

Originally Posted by MrOats

For the second question:
I got Amplitude = 4, Period = [2(pie)/b], Frequency = 2/4
...Not so sure about this one...

And for the equation: y= 4 Sin (2/4(x)+4)

...I really am unsure about this one. Someone please clarify for me? I need to get "solid" in my head what the amplitude, frequency, period, and range are...And solid, a=, b=, c=...answers.

Please help me demystify this sin wave question! Thank you.

I think you are misinterpreting the question ....

$\displaystyle y = (a\sin bx) + c$
Is the distance x from the wall or the person?

Anyway I will assume from the wall,
Note that when x = 0, then y = 3. So c = 3.

Since the range of variation for y is c-a to c+a (because $\displaystyle \sin bx$ varies from -1 to 1) and the question tells us the ropes height is from 1 to 5 feet, we have $\displaystyle 3 - a =1$ and $\displaystyle 3 + a = 5$. This gives us $\displaystyle a = 2$

$\displaystyle 2\sin bx + 3$ has the same period as $\displaystyle \sin bx$. Now since the period of a sin function is $\displaystyle 2\pi$, we are forced to choose $\displaystyle b = \frac12$.

Thus the final equation is
$\displaystyle y = 2\sin \frac{x}{2} + 3$

Side Note:Now since $\displaystyle 4\pi$ is the period, it means $\displaystyle a\sin bx + c = a\sin b(x+4\pi) + c \Rightarrow bx+4\pi b = n\pi+ (-1)^n bx$. Clearly for n=2, the equation reads $\displaystyle 4\pi b = 2\pi$ and thus b=2

Thank you very much Isomorphism!

I get it now. Finally. =)