If $8\tan x=15$ then $\sin x-\cos x$ is equal to
a)$\dfrac{8}{17}$
b)$\dfrac{17}{7}$
c)$\dfrac{1}{17}$
d)$\dfrac{7}{17}$
VerifiedHint: Here, first find the value of $\tan x$ from the expression $8\tan x=15$. We have to apply the definition, $\tan x=\dfrac{opposite\text{ }side}{adjacent\text{ }side}$. Then, construct a right triangle. With the help of the right triangle, find the hypotenuse using Pythagoras theorem. Next, find the values of $\sin x$ and $\cos x$, substitute these values in the expression $\sin x-\cos x$.
Complete step-by-step answer:
Here, we are given that $8\tan x=15$.
Now, we have to find the value of $\sin x-\cos x$.
First consider,
$8\tan x=15$
Now, by cross multiplication we get:
$\tan x=\dfrac{15}{8}$
Now, consider the figure,
We know that,
$\tan x=\dfrac{opposite\text{ }side}{adjacent\text{ }side}$
In the figure we have,
Opposite side = AC
Adjacent side = AB
Hypotenuse = BC
$\Delta ABC$ is a right angled triangle. Hence, we can apply the Pythagoras theorem.
Now, by Pythagoras theorem we have,
$ {{(Hypotenuse)}^{2}}={{(Opposite\text{ }side)}^{2}}+{{(Adjacent\text{ }side)}^{2}} $
$\Rightarrow {{(BC)}^{2}}={{(AC)}^{2}}+{{(AB)}^{2}} $
Here, we have,
$\tan x=\dfrac{AC}{AB}$
AC = 15
AB = 8
Now, we can write:
$ {{(BC)}^{2}}={{15}^{2}}+{{8}^{2}} $
$ \Rightarrow {{(BC)}^{2}}=225+64 $
$ \Rightarrow {{(BC)}^{2}}=289 $
Next, by taking square root on both the sides we get,
$BC=\sqrt{289} $
$ \Rightarrow BC=17 $
We know that,
$ \sin x=\dfrac{Opposite\text{ }side}{Hypotenuse} $
$ \Rightarrow \sin x=\dfrac{AC}{BC} $
$ \Rightarrow \sin x=\dfrac{15}{17} $
Similarly, we have,
$ \cos x=\dfrac{\text{Adjacent }side}{Hypotenuse} $$ \Rightarrow \cos x=\dfrac{AB}{BC} $
$ \Rightarrow \cos x=\dfrac{8}{17} $
Now, we have to find $\sin x-\cos x$.
By substituting the values of $\sin x$ and $\cos x$ in the above expression we get,
$\Rightarrow \sin x-\cos x=\dfrac{15}{17}-\dfrac{8}{17}$
Next, by taking LCM we obtain:
$ \Rightarrow \sin x-\cos x=\dfrac{15-8}{17} $
$ \Rightarrow \sin x-\cos x=\dfrac{7}{17} $
Therefore, we can say that when $8\tan x=15$, then $\sin x-\cos x=\dfrac{7}{17}$.
Hence, the correct answer for this question is option (d).
Note: Here, you must have an idea about the trigonometric ratios and the definitions. If you don’t have the basic idea it will be very difficult to continue the solution. To get a better idea, always construct a right triangle and mark the angle given, and it’s corresponding opposite side, adjacent side and hypotenuse.
