Question

If tan A = 3/5, find the value of sec A + cosec A

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:\sec\:A\:+\:cosec\:A\:=\:\dfrac{8\:\sqrt{34}}{15}}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\tan\:A\:=\:\dfrac{3}{5}}$

We have to find the value of

$\displaystyle{\sf\:\sec\:A\:+\:cosec\:A}$

Now,

$\displaystyle{\sf\:\tan\:A\:=\:\dfrac{3}{5}}$

$\displaystyle{\implies\sf\:\cot\:A\:=\:\dfrac{1}{\tan\:A}}$

$\displaystyle{\implies\sf\:\cot\:A\:=\:\dfrac{5}{3}}$

Now, we know that,

$\displaystyle{\pink{\sf\:1\:+\:\tan^2\:A\:=\:\sec^2\:A}}$

$\displaystyle{\implies\sf\:\sec^2\:A\:=\:1\:+\:\left(\:\dfrac{3}{5}\:\right)^2}$

$\displaystyle{\implies\sf\:\sec^2\:A\:=\:1\:+\:\dfrac{9}{25}}$

$\displaystyle{\implies\sf\:\sec^2\:A\:=\:\dfrac{25\:+\:9}{25}}$

$\displaystyle{\implies\sf\:\sec^2\:A\:=\:\dfrac{34}{25}}$

$\displaystyle{\implies\boxed{\green{\sf\:\sec\:A\:=\:\dfrac{\sqrt{34}}{5}}}}$

Now, we know that,

$\displaystyle{\pink{\sf\:1\:+\:\cot^2\:A\:=\:cosec^2\:A}}$

$\displaystyle{\implies\sf\:cosec^2\:A\:=\:1\:+\:\left(\:\dfrac{5}{3}\:\right)^2}$

$\displaystyle{\implies\sf\:cosec^2\:A\:=\:1\:+\:\dfrac{25}{9}}$

$\displaystyle{\implies\sf\:cosec^2\:A\:=\:\dfrac{25\:+\:9}{9}}$

$\displaystyle{\implies\sf\:cosec^2\:A\:=\:\dfrac{34}{9}}$

$\displaystyle{\implies\boxed{\blue{\sf\:cosec\:A\:=\:\dfrac{\sqrt{34}}{3}}}}$

Now, we have to find the value of

$\displaystyle{\sf\:\sec\:A\:+\:cosec\:A}$

$\displaystyle{\implies\sf\:\dfrac{\sqrt{34}}{5}\:+\:\dfrac{\sqrt{34}}{3}}$

$\displaystyle{\implies\sf\:\dfrac{3\:\sqrt{34}\:+\:5\:\sqrt{34}}{5\:\times\:3}}$

$\displaystyle{\implies\sf\:\dfrac{8\:\sqrt{34}}{15}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\sec\:A\:+\:cosec\:A\:=\:\dfrac{8\:\sqrt{34}}{15}}}}}$

Answer 2

Answer :

( sec A + cosec A = 8√34 / 15 )

Explanation :

Given,

tan A = 3/5

To find,

sec A + cosec A

Now,

tan A = 3/5

=> cot A = 1 / tan A

=> cot A = 5 / 3

We know,

1 + tan2 A = sec2 A

=> sec2 A = 1 + (3/5) 2

=> sec2 A = 1 + 9/25

=> sec2 A = 25+9 / 25

=> sec2 A = 34 / 25

=> sec A = √34 / 25

And we know,

1 + cot2 A = cosec2 A

cosec2 A = 1 + (5 / 3)2

cosec2 A = 1 (25 / 9)

cosec2 A = (25 + 9 / 9)

cosec2 A = (34 / 9)

cosec A = (34 / 3)

Now Find the value,

sec A + cosec A

(34 / 5) + (34 / 3)

= 3√34 + 5√34 / 5×3

= 8√34 / 15

sec A + cosec A = 8√34 / 15