The value of sin(2x) is [tex]\frac{120}{169}[/tex]
Explanation:
Given that [tex]tan x =\frac{12}{5}[/tex]
The formula for [tex]sin(2x)[/tex] is [tex]\sin (2 x)=2 \sin x \cos x[/tex]
Since, [tex]\tan x=\frac{o p p}{a d j}[/tex]
Also, it is given that [tex]tan x =\frac{12}{5}[/tex]
Thus, [tex]opp=12[/tex] and [tex]adj=5[/tex]
To find the hypotenuse, let us use the pythagoras theorem,
[tex]\begin{aligned}h y p &=\sqrt{12^{2}+5^{2}} \\&=\sqrt{144+25} \\&=\sqrt{169} \\&=13\end{aligned}[/tex]
Now, we can find the value of sin x and cos x.
[tex]\sin x=\frac{\text { opp }}{h y p}=\frac{12}{13}[/tex]
[tex]\cos x=\frac{a d j}{h y p}=\frac{5}{13}[/tex]
Now, substituting these values in the formula for sin 2x, we get,
[tex]\begin{aligned}\sin (2 x) &=2 \sin x \cos x \\&=2\left(\frac{12}{13}\right)\left(\frac{5}{13}\right) \\&=\frac{120}{169}\end{aligned}[/tex]
Thus, the value of sin(2x) is [tex]\frac{120}{169}[/tex]
