Solve for x:
sec(2x)csc(2x) = 2csc(x)
for 0 < x < 2pi
[the first "<" sign is less than or equal to]
Thank you!
sec(2x)csc(2x) = 2csc(x)
1/((cos 2x)(sin 2x)) = 2/sinx
cross-multiply
2(cos 2x)(sin 2x) = sinx
2(cos 2x)(2sinxcosx) = sinx
divide by sinx
2(cos 2x)(2cosx) = 1
4cosx(2cos^2 x - 1) = 1
8cos^3 x - 4cosx - 1 = 0
let's let cosx = a
so we are solving
8a^3 - 4a - 1 = 0
after a few tries I go a = -1/2 to work
giving me
(2a-1)(4a^2 - 2a - 1) = 0
a = -.5 or a = .809 or -.309
so cosx = -.5 or cosx = .809 or -.309
I will do one of these:
cosx = -.5, the reference angle is pi/3 or 60º
but the cosine is negative in quadrants II and III
so x = pi - pi/3 = 2pi/3 or 180-60 = 120º
or x = pi + pi/3 = 4pi/3 or 180+60 = 240º
You should have 6 different answers
Left hand side -
sec(2x)csc(2x)= 1/ cos(2x) 1/sin(2x)
= 1/(Cos2x)* 1/(2 Sinx Cosx)
Right Hand side = 1/2Sinx
Equating LHS & RHS
1/{Cos2x}* 1/(2 Sinx Cosx)
= 1/2Sinx
or
1/{Cos2x}* 1/cos x = 1
or {Cos2x}* Cosx = 1
this actually only holds true for x = 0 and x = 2pi
Vipster, if x = 0, the original equation has undefined calculations, csc 0 is undefined.
you have an error by saying
2csc(x) = 1/2Sinx
2csc(x) = 2/sinx
To solve the equation sec(2x)csc(2x) = 2csc(x), the first step is to simplify the equation using trigonometric identities.
First, let's recall some basic trigonometric identities:
1. sec(x) = 1/cos(x)
2. csc(x) = 1/sin(x)
Now, let's rewrite the equation using these identities:
(1/cos(2x))(1/sin(2x)) = 2(1/sin(x))
Next, we can simplify the left side of the equation by multiplying the denominators together and the numerators together:
1 / (sin(2x) cos(2x)) = 2 / sin(x)
To simplify the right side of the equation, we can multiply both sides by sin(x):
1 / (sin(2x) cos(2x)) * sin(x) = 2
Now, we can simplify the left side of the equation:
1 / (2sin(x) cos(x) cos(2x)) = 2
Next, we'll multiply both sides of the equation by 2sin(x) cos(x) cos(2x):
1 = 4sin(x) cos(x) cos(2x)
Now, we need to use the double angle identity for cosine:
cos(2x) = 2cos^2(x) - 1
Substituting this into the equation:
1 = 4sin(x) cos(x) (2cos^2(x) - 1)
Simplifying further:
1 = 8sin(x) cos^3(x) - 4sin(x) cos(x)
Now, we can substitute sin(x) = 1/csc(x) and cos(x) = 1/sec(x) into the equation:
1 = 8 (1/csc(x)) (1/sec(x))^3 - 4 (1/csc(x)) (1/sec(x))
To simplify:
1 = 8 / (csc(x) sec^3(x)) - 4 / (csc(x) sec(x))
Now, we'll multiply both sides by csc(x) sec(x) to get rid of the denominators:
(csc(x) sec(x)) = 8 / (csc(x) sec^3(x)) - 4 / (csc(x) sec(x))
Simplifying further, we can combine the two fractions on the right side:
(csc(x) sec(x)) = (8 - 4 sec^2(x)) / (csc(x) sec^3(x))
Now, we can cancel out the csc(x) and sec(x) on both sides of the equation:
1 = 8 - 4 sec^2(x)
Rearranging the equation:
4 sec^2(x) = 7
Finally, divide both sides by 4:
sec^2(x) = 7/4
Taking the square root of both sides (don't forget about the ± sign):
sec(x) = ±√(7/4)
To find the possible values for x, we need to consider the range 0 < x < 2π, which means that x is greater than 0 and less than 2π.
Using the inverse function of sec(x), which is arcsec(x), we can find the values of x:
x = arcsec(±√(7/4))
Depending on the values of arcsec(±√(7/4)), we can determine the solutions within the given range of 0 < x < 2π.
Note: The solutions may be further simplified or approximated depending on the level of precision required.