How do you find the exact value of sec ⁡ ( tan − 1 ⁡ ( − 3 5 ) )?

Let θ = tan − 1 ⁡ ( − 3 5 )then tan ⁡ θ = − 3 5 and sec 2 ⁡ θ = 1 + tan 2 ⁡ θ = 1 + 9 25 = 34 25 As tan ⁡ θ is negative, as range of tan − 1 is − π 2 &lt; θ &lt; π 2 , θ lies in Q4 and sec ⁡ θ ispositive and sec ⁡ θ = 34 5 and hence sec ⁡ ( tan − 1 ⁡ ( − 3 5 ) ) = sec ⁡ θ = 34 5

Explained: "How do you find the exact value of..."

Adison Rogers

Answered question

2022-11-23

How do you find the exact value of

\sec (\tan^{- 1} (- \frac{3}{5}))

Answer & Explanation

Neil Short

Beginner2022-11-24Added 17 answers

Let

θ = \tan^{- 1} (- \frac{3}{5})

then

\tan θ = - \frac{3}{5}

and

\sec^{2} θ = 1 + \tan^{2} θ = 1 + \frac{9}{25} = \frac{34}{25}

\tan θ

is negative, as range of

\tan^{- 1}

- \frac{π}{2} < θ < \frac{π}{2}, θ

lies in Q4 and

\sec θ

is
positive and

\sec θ = \frac{\sqrt{34}}{5}

and hence

\sec (\tan^{- 1} (- \frac{3}{5}))

= \sec θ

= \frac{\sqrt{34}}{5}

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