Question

Given that tan x =0.4 and x is an acute angle,find the exact value of sin 2x

Answer 1

I think there is an identity relating \(\sin(2x)\) with \(\tan(x)\) as :
\[\sin(2x) = \frac{2 \tan(x)}{1 + \tan^2(x)}\]

Answer 2

As:
\[\tan(x) = 0.4 \implies \tan(x) = \frac{2}{5}\]

Answer 3

I have converted that decimal given value to fraction, so you can just use the decimal value too if you have calculator with you. :)

Answer 4

So, using that formula:
\[\large \sin(2x) = \frac{2 \times \frac{2}{5}}{1 + (\frac{2}{5})^2} \implies \sin(2x) = \frac{\frac{4}{5}}{\frac{29}{25}}\]

Answer 5

\[\sin(2x) = \frac{4}{5} \times \frac{25}{29} \implies \sin(2x) = \frac{20}{29}\]

Answer 6

you can also convert the final result into decimals if you want. :)

Answer 7

The more common identity to know is sin(2x) = 2sin(x)cos(x). It is a slower method, but it's an identity you'll likely be shown and be forced to remember. This will get you the same answer, but this is, what I believe, the more common way of having to do this problem:
Since tanx = 0.4 and we know tanx gives the opposite and adjacent sides of a triangle. we can rewrite 0.4 as 2/5 and name 2 as our opposite length and 5 as our adjacent length.



The hypotenuse was required, so I did pythagorean theorem to get that missing side. From there, we can then gather that \(\sin x = \frac{2}{\sqrt{29}}\) and \(\cos x = \frac{5}{\sqrt{29}}\). Plugging those into the identity, we have:

\[2(\frac{ 2 }{ \sqrt{29} })(\frac{ 5 }{ \sqrt{29} }) = \frac{ 20 }{ 29 }\]