CPhill

I've not run across the term "surd" before......Simply put, it is a number that cannot be simpified to remove a root (an irrational number is the result).....

How did we get the word "Surd" ?

Well around 820 AD al-Khwarizmi (the Persian guy who we get the name "Algorithm" from) called irrational numbers "'inaudible" ... this was later translated to the Latin surdus ("deaf" or "mute")

BTW......Thanks Aziz, for that answer........

I believe the questioner wants to take the square root of 69......we cannot "reduce" (simplify) this any further, since the only factors of 69 are 23 and 3, and these are both prime..

Aziz's answer is correct..I mistook "loge" as a mispelling of "log"...but I see now that the intent was "ln".....

Rearranging  a little, we have...

4x^2 - 7x - 2 = 0

And factoring produces

(4x + 1 ) (x - 2) = 0

And setting each factor to 0, we obtain that

x = -1/4  and   x  =  2   as Aziz has said...

log(.25) = -0.602

Note that sec⁶x can be written as   [sec²x]³ = [(1 + tan²x)]³

And using the binomial theorem, we have.........

[(1 + tan²x)]³  = [tan²x]³ + 3[tan²x]² + 3[tan²x] + 1    ...   therefore  ....

sec⁶x - tan⁶x =

 [tan²x]³ + 3[tan²x]² + 3[tan²x] + 1  - tan⁶x   =

 tan⁶x  + 3[tan²x]² + 3[tan²x] + 1 - tan⁶x   =

3[tan⁴x] + 3[tan²x] + 1   = which is the same thing as the RHS.....

From a trig identity,

cos(Θ) = cos (-Θ)  therefore

cos (wt-x) = cos(-(wt-x)) = cos(x-wt)      so we have ..  (using the angle difference indentity for the cosine)

Acos(x-wt)  = A[cos(x)cos(wt) + sin(x)sin(wt)]  = Asin(x)sin(wt) + Acos(x)cos(wt)    ....and that = the left hand side 

Not really a math question, but hey, we're a full service garage....read all about it here.....

Yeah...that Human Organ one is the best.........

Melody, I used WolframAlpha...just type in x>3...you will see the graph displayed...then you can do a "snip" and upload it.....