Question

If tan9x=3/4 then 3cosec3x-4sec3x=?

Answer 1

The value of 3cosec(3x) - 4sec(3x) = 9.924

Step-by-step explanation:

$\textbf{\large Let 3x = y}\\\\\textbf{\large So, }\tan (9x) = \frac{3}{4} = \tan (3y)\\ \\\\So, \sin (3y) = \frac{3}{5} and$

and $\cos (3y) = \frac{4}{5}$                                               (equation 1 )

(because here by the information of tan(3y)  is that its height = 3 and base = 4)

So

$\textbf{\large By the Pythagorus theorem}\\\\base^2 + height ^2 = altitute^2\\\\4^2 + 3^2 = altitute^2 = 25\\So, altitute = 5$

By equation 1,   Taking inverse of sin on both sides.

3y = $\sin^-^1 \frac{3}{5} = 0.643 radian$

so,

y = 0.214 radian

As we know that y = 3x

So, putting the value of 3x in given equation

$3\times cosec(3x) - 4\times sec (3x)=\\\\3\times cosec(0.214) - 4\times sec (0.214)=$                (equation 2)

Now as ,

cosec(0.214) = 4.672

and

sec(0.214) = 1.023

putting these values in equation 2.

$3\times (4.672) - 4\times (1.023) = \textbf{\Large 9.924}\\\\\textbf{\Large So the value of 3cosec(3x) - 4sec(3x) = 9.924 }$