TRIG HELP PLEASE!

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Im having trouble with these problems!

x=theta
solve the equation on the interval [0,2 pi)
1.
sin(2x)+sin x=0



2.
x=theta
csc^5(x) - 4csc x=0



3.
A mass hangs from a spring which oscillates up and down. The position P (in feet) of the mass at time T (in seconds) is given by:
P=4cos(4T)
For what values of T, [0,pi), will the position be 2sq. root of 2 feet? find the exact values.



4.
From the edge of a 1000-foot cliff, the angles of deppresion to two cars in the valley below are 21 degrees and 28 degrees. How far apart are the cars?



5.
A weight is suspended on a system of spring and oscillates up and down according to :
P= 0.1[3cos(8T) - sin(8T)]
where P is the position in meters above or below the point of of equilibrium (P=0) and T is time in seconds. Find the time when the weight is at equillibrium. Find all the values of T [0,1], rounded to the nearest 0.01 second. Give answer in radians.
 
Hello, brooke!

Here are the first two . . .

Solve the equation on the interval \(\displaystyle [0,2\pi)\)

\(\displaystyle 1)\;\;\sin(2\theta)\,+\,\sin(\theta)\;=\;0\)
Click to expand...
From the identity: .\(\displaystyle \sin(2x)\:=\:2\cdot\sin(x)\cdot\cos(x)\)

. . we have: . \(\displaystyle 2\cdot\sin(\theta)\cdot\cos(\theta)\,+\,\sin(\theta)\;=\;0\)


Factor: .\(\displaystyle \sin(\theta)\cdot[2\cdot\cos(\theta)\,+\,1]\;=\;0\)


We have two equations to solve:

. . \(\displaystyle \sin(\theta)\,=\,0\;\;\Rightarrow\;\;\theta\,=\,0,\,\pi\)

. . \(\displaystyle 2\cdot\cos(\theta)\,+\,1\:=\:0\;\;\Rightarrow\;\;\cos(\theta)\,=\,-\frac{1}{2}\;\;\Rightarrow\;\;\theta\,=\,\frac{2\pi}{3},\,\frac{4\pi}{3}\)


\(\displaystyle 2)\;\;\csc^5(\theta)\,-\,4\csc(\theta)\;=\;0\)
Click to expand...
Factor: . \(\displaystyle \csc(\theta)\cdot[\csc^4(\theta)\,-\,4]\;=\;0\)

Factor: . \(\displaystyle \csc(\theta)\cdot[\csc^2(\theta)\,-\,2][\csc^2(\theta)\,+\,2]\;=\;0\)


We have three equation to solve:

. . \(\displaystyle \csc(\theta)\,=\,0\;\;\Rightarrow\;\;\theta\,=\,0,\,\pi\)

. . \(\displaystyle \csc^2(\theta)\,-\,2\:=\:0\;\;\Rightarrow\;\;\csc(\theta)\,=\,\pm\sqrt{2}\;\;\Rightarrow\;\;\theta\,=\,\frac{\pi}{4},\,\frac{3\pi}{4},\,\frac{5\pi}{4},\;\frac{7\pi}{4}\)

. . \(\displaystyle \csc^2(\theta)\,+\,2\:=\:0\;\;\Rightarrow\;\;\csc^2(\theta)\,=\,-2\;\;\Rightarrow\;\;\) no real soltuions.
 
PLEASE HELP WITH TRIG

Im still having trouble with these 3. can someone please help?

3.
A mass hangs from a spring which oscillates up and down. The position P (in feet) of the mass at time T (in seconds) is given by:
P=4cos(4T)
For what values of T, [0,pi), will the position be 2sq. root of 2 feet? find the exact values.



4.
From the edge of a 1000-foot cliff, the angles of deppresion to two cars in the valley below are 21 degrees and 28 degrees. How far apart are the cars?



5.
A weight is suspended on a system of spring and oscillates up and down according to :
P= 0.1[3cos(8T) - sin(8T)]
where P is the position in meters above or below the point of of equilibrium (P=0) and T is time in seconds. Find the time when the weight is at equillibrium. Find all the values of T [0,1], rounded to the nearest 0.01 second. Give answer in radians.
 
3) Plug in the given value, 2sqrt(2), for "P", and solve. (This works like the problems that have already been completed for you.)

4) Draw the two nested right triangles. Label the angles with the given values. Label the heights as "1000". Find the lengths of the two bases. Subtract them to find how far apart they are.

5) Hint: What is the meaning of "equilibrium" in relation to "position"? Then solve, as show in the two exercises already completed for you.

If you get stuck, please reply showing what you have tried. Thank you.

Eliz.
 
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