Maths Genuises!!! Please help!?

5sin(2x)=2cos(2x), x: 0 to 360°
tan(2x)=2/5
2x=arctan(2/5)≈ 21.8°
x≈10.9°, 100.9°, 190.9°, 280.9°

5sin2x=2cos2x

sin2x/cos2x = 2/5
tan2x = 2/5
2x = tan-1(2/5) = 21.8
x= 10.9 in the first circle
x: 0 to 360 deg
(x = 10.9)
x= 100.9 in the 2nd circle ( 10.9 + 90)
x=190.9 in the 3rd circle (10.9 + 2(90))
x= 280.9 in the 4th circle (10.9 + 3(90))

(The difference is 90 degrees)

Here you go :)

Sorry I just looked at your answer.
Your answer is in degree modes. Make sure your familiar with the mode your calculator gives, it makes a huge difference.

k=integers
in degree
(arctan(2/5)+180*k)/2 should give you 10.0,100.9,190.9,280.9,370.9, so on.

Hope this answers all your question you may have.

　
5 sin(2x) = 2 cos(2x)
sin(2x)/cos(2x) = 2/5
tan(2x) = 2/5
2x = tan⁻¹(2/5) + 180k, for any integer k
x = 1/2 tan⁻¹(2/5) + 90k
x = 10.9° + 90k

For x on interval [0°, 360°) we get

x = 10.9°, 100.9°, 190..9°, 280.9°

5sin2x = 2cos2x

so, sin2x/cos2x = 2/5

=> tan2x = 2/5

i.e. 2x = 21.8° + 180n°...for n ∈ ℤ

so, x = 10.9° + 90n°

Hence, x = 10.9°, 10.9 + 90, 10.9 + 2(90), 10.9 + 3(90),...

=> x = 10.9°, 100.9°, 190.9° and 280.9°...for 0 < x < 360°

:)>

Do a little rearranging (divide from both sides) to get:

[sin2x] / [cos2x] = 2/5

The left side is equivalent to tan2x (because tan = sin/cos):

tan2x = 2/5

Take the inverse tan of 2/5 using a calculator:

The calculator will give you: 21.8014...

Because it's the tangent function, the angle directly across it on the unit circle has the same sin/cos ratio, so in addition to 21.8, we can add 180, meaning tan(2/5) also = 201.8. Inverse functions are just weird that way, and if it doesn't make sense you should memorize the +180 rule for inverse tan (it's different for inverse sin and cos). Then, because it's a circle you can add 360 degrees to either 21.8 or 201.8 an infinite number of times and it'll be the same angle. So:

2x = 21.8, 201.8, 381.8, 561.8, etc....

Divide both sides by 2 and x = 10.9, 100.9, 190.9, 280.9, etc....