Question

If cos thita =5/7 find the value of tan thita + cosec thita
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Answer 1

Given :-

${cos\theta} = \dfrac{5}{7}$

To find :-

${tan\theta + cosec\theta}$

SOLUTION:-

As we know that ,

${cos\theta} = \dfrac{adjacent}{hypotenuse}$

Since,

Adjacent side is 5

Hypotenuse is 7

As we also know that ,

Pythagoras theorem :-

$(opp) {}^{2} + (adj) {}^{2} = (hyp) {}^{2}$

From this we shall find opposite side

$(opp) {}^{2} + (5) {}^{2} = (7) {}^{2}$

$(opp) {}^{2} + 25 = 49$

$(opp) {}^{2} = 49 - 25$

$(opp) {}^{2} = 24$

$opp = \sqrt{24}$

So,

${tan\theta} = \dfrac{opposite}{adjacent}$

${cosec\theta} = \dfrac{hypotenuse}{opposite}$

${tan\theta} = \dfrac{ \sqrt{24} }{5}$

${cosec\theta} = \dfrac{7}{ \sqrt{24} }$

${tan\theta+cosec\theta} = \dfrac{ \sqrt{24} }{5} + \dfrac{7}{ \sqrt{24}}$

$= \dfrac{ \sqrt{24} }{5} + \dfrac{7}{ \sqrt{24} }$

$= \dfrac{24 + 35}{5 \sqrt{24} }$

$= \dfrac{59}{5 \sqrt{24} }$

$= \dfrac{59}{10 \sqrt{6} }$

Know more :-

Trigonometric Identities:-

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

csc²θ - cot²θ = 1

Trigonometric relations:-

sinθ = 1/cscθ

cosθ = 1 /secθ

tanθ = 1/cotθ

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Trigonometric ratios:-

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

cotθ = adj/opp

cscθ = hyp/opp

secθ = hyp/adj