Answer:
Hence, the value of tan2x is:
24/7
Step-by-step explanation:
We are given sine trignometric ratio as:
[tex]\sin x=\dfrac{4}{5}[/tex]
Aklso the angle ''x'' lie in the second quadrant.
As we know that the trignometric ratios which are positive in second quadrant are:
Sine(sin) and cosecant(csc)
whereas the other trignometric ratio's i.e. cosine(cos),secant(sec),cotangent(cot),tangent(tan) are all negative in second quadrant.
We know that the sine trignometric function is the ratio of perpendicular to hypotenuse of a right angled triangle corresponding to angle 'x'.
i.e. Let P=4 and H=5
As we know that in a right triangle we have:
[tex]H^2=P^2+B^2\\\\5^2=4^2+B^2\\\\25=16+B^2\\\\B^2=25-16\\\\B^2=9\\\\B=3[/tex]
Also as we know,
[tex]\tan x=\dfrac{P}{B}\\\\i.e.\\\\\tan x=-\dfrac{4}{3}[/tex]
(The value is negative as tangent is negative in second quadrant)
Also, we are asked to find the value of: [tex]\tan 2x[/tex]
We know that:
[tex]\tan 2x=\dfrac{2\tan x}{1-\tan^2 x}\\\\\\Hence,\\\\\\\tan 2x=\dfrac{2\times \dfrac{-4}{3}}{1-(\dfrac{-4}{3})^2}\\\\\\\tan 2x=\dfrac{\dfrac{-8}{3}}{1-\dfrac{16}{9}}\\\\\\\tan 2x=\dfrac{\dfrac{-8}{3}}{\dfrac{9-16}{9}}\\\\\\\tan 2x=\dfrac{\dfrac{-8}{3}}{\dfrac{-7}{9}}\\\\\\\tan 2x=\dfrac{-24}{-7}\\\\\\\tan 2x=\dfrac{24}{7}[/tex]
