How do you find the exact value of $tan (arcsin (\frac{1}{3}))$ $tan(arcsin(1/3))$ ?

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How do you find the exact value of $tan (arcsin (\frac{1}{3}))$ $tan(arcsin(1/3))$ ?

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Answer 1 · 2017-02-09T16:01:51

$tan (arcsin (\frac{1}{3})) = \frac{\sqrt{2}}{4}$ $tan(arcsin(1/3)) = sqrt(2)/4$

Explanation:

Consider a right angled triangle with sides $1$ $1$ , $2 \sqrt{2}$ $2sqrt(2)$ and $3$ $3$

We can tell that it is right angled since:

$1^{2} + {(2 \sqrt{2})}^{2} = 1 + 8 = 9 = 3^{2}$ $1^2+(2sqrt(2))^2 = 1+8 = 9 = 3^2$

Denote the smallest internal angle by $θ$ $theta$ .

Then:

$sin (θ) = \frac{opposite}{hypotenuse} = \frac{1}{3}$ $sin(theta) = "opposite"/"hypotenuse" = 1/3$

$tan (θ) = \frac{opposite}{adjacent} = \frac{1}{2 \sqrt{2}} = \frac{\sqrt{2}}{4}$ $tan(theta) = "opposite"/"adjacent" = 1/(2sqrt(2)) = sqrt(2)/4$

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How do you find the exact value of $tan (arcsin (\frac{1}{3}))$ $tan(arcsin(1/3))$ ?

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