The value of \(\tan(2\theta)\) is \(\frac{4\sqrt 2}{7}\)
The given parameter is:
\(\sin(\theta) = \frac 13\)
Calculate cos(theta) using the following trigonometry ratio
\(\cos^2(\theta) = 1 - \sin^2(\theta)\)
So, we have:
\(\cos^2(\theta) = 1 - (1/3)^2\)
\(\cos^2(\theta) = 1 - \frac 19\)
Subtract
\(\cos^2(\theta) = \frac 89\)
Take the square roots of both sides
\(\cos(\theta) = \frac{2}{3}\sqrt 2\)
tan (2 theta) is then calculated as:
\(\tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)}\)
This gives
\(\tan(2\theta) = \frac{2\sin(\theta)\cos(\theta)}{\cos^2(\theta) - \sin^2(\theta)}\)
So, we have:
\(\tan(2\theta) = \frac{2 * 1/3 * 2/3 \sqrt 2}{8/9 - 1/9}\)
Simplify
\(\tan(2\theta) = \frac{4/9 \sqrt 2}{7/9}\)
Further, simplify
\(\tan(2\theta) = \frac{4\sqrt 2}{7}\)
Hence, the value of \(\tan(2\theta)\) is \(\frac{4\sqrt 2}{7}\)
Read more about trigonometry ratios at:
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