How do you evaluate ${tan}^{- 1} (0)$ $tan^-1(0)$ without a calculator?

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How do you evaluate ${tan}^{- 1} (0)$ $tan^-1(0)$ without a calculator?

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Answer 1 · 2016-12-01T22:49:02

$θ = 0$ $theta=0$

Explanation:

Find ${tan}^{- 1} (0)$ $tan^-1 (0)$ without a calculator.

${tan}^{- 1} (0)$ $tan^-1(0)$ refers to the ANGLE whose tangent equals zero.

Recall the identity $tan θ = \frac{sin θ}{cos θ}$ $tantheta = sintheta/costheta$

If $tan θ = 0$ $tantheta=0$ , then the numerator of $\frac{sin θ}{cos θ}$ $sintheta/costheta$ must also equal zero.

So, $sin θ = 0$ $sintheta"=0$ .

Referring to the unit circle, the angles with a sine of zero are
$θ = 0, π$ $theta =0, pi$ .

BUT, IT'S A BIT MORE COMPLICATED!

The range of ${tan}^{- 1}$ $tan^-1$ (also called arctan) is $- \frac{π}{2}$ $-pi/2$ to $\frac{π}{2}$ $pi/2$ .

In other words, the only "allowed" values of ${tan}^{- 1}$ $tan^-1$ fall within this range. So, the the answer $θ = π$ $theta=pi$ is not allowed, and the answer to the problem ${tan}^{- 1} (0)$ $tan^-1(0)$ is $θ = 0$ $theta=0$

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How do you evaluate ${tan}^{- 1} (0)$ $tan^-1(0)$ without a calculator?

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How do you evaluate ${tan}^{- 1} (0)$ $tan^-1(0)$ without a calculator?