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C2 Trig Help!

Solve for t (theta), giving all roots in the interval 0 =< t =< 360:

1) 4sin^2 t cos t = tan^2 t.

Right so I started by doing the following:

4 - 4cos^2 cos t = sin^2 t / cos^2 t,
cos^2 t (4 - 4cos^2 cos t) = 1 - cos^2 t,
cos^2 t (4 - 4cos^3 t) = 1 - cos^2 t,
4 cos^2 t - 4 cos^5 t = 1 - cos^2 t,
Then use x = cos t to do:
4x^2 - 4x^5 = 1 - x^2,
= 5x^2 - 4x^5 - 1 = 0.

Is this right? If so, how would I factorise?
If it is incorrect, what is the correct way to do this?

Many thanks :smile:

Reply 1

Quadratic

Not sure if this is the best way either, but I found a more manageable solution.

Try dividing by the Sin^2(t) in the first equation and see where it takes you.

Reply 2

adil12

Original post by Quadratic

Not sure if this is the best way either, but I found a more manageable solution.

Try dividing by the Sin^2(t) in the first equation and see where it takes you.

Surely that would make it more difficult? And can you divide by sin^2 t in this case? Our teacher says it eliminates a solution or something... :s-smilie:

Reply 3

Quadratic

Original post by adil12

Surely that would make it more difficult? And can you divide by sin^2 t in this case? Our teacher says it eliminates a solution or something... :s-smilie:

Ah damn yeah, never thought about that. I have a habit of dividing by 0. 2 min.

Reply 4

jaheen22

x = 1 is a solution to the equation.

Reply 5

adil12

Original post by jaheen22

x = 1 is a solution to the equation.

How did you work it out?

Reply 6

jaheen22

Original post by adil12

How did you work it out?

Just by looking at it:

5(1)^2 - 4(1)^5 - 1 = 0

Reply 7

adil12

Anybody know if my method is correct and if so, how to factorise it to find the solutions?

Reply 8

Murrayland

4sin^2(t)cos(t) = tan^2(t)
4sin^2(t)cos(t) = sin^2(t) / cos^2(t)
4sin^2(t)cos^3(t) = sin^2(t)
sin^2(t)[4cos^3(t) - 1] = 0

t = sin^-1(0) OR t = cos^-1((1/4)^(1/3))
t = ... :smile:

Reply 9

jaheen22

Original post by adil12

How did you work it out?

The equation is wrong though I think it should be:

4x^5 - 4x^3 - x^2 + 1 = 0

Where x = 1 is still a solution.

Reply 10

Quadratic

Original post by Murrayland

4sin^2(t)cos(t) = tan^2(t)
4sin^2(t)cos(t) = sin^2(t) / cos^2(t)
4sin^2(t)cos^3(t) = sin^2(t)
sin^2(t)[4cos^3(t) - 1] = 0

t = sin^-1(0) OR t = cos^-1((1/4)^(1/3))
t = ... :smile:

Cheers aha, this was bugging me. Seems I can't do C2 any more :P

Reply 11

adil12

Original post by Murrayland

4sin^2(t)cos(t) = tan^2(t)
4sin^2(t)cos(t) = sin^2(t) / cos^2(t)
4sin^2(t)cos^3(t) = sin^2(t)
sin^2(t)[4cos^3(t) - 1] = 0

t = sin^-1(0) OR t = cos^-1((1/4)^(1/3))
t = ... :smile:

Thanks a lot mate! Repped :biggrin: