Question

If cos theta=t(0<t<1) and theta does not lies in the first quadrant find the values of sin theta and Tan theta​

Answer 1

SOLUTION

GIVEN

$\sf{ \cos \theta \: = t \: \: \: \: \: where \: \: 0 < t < 1}$

and θ does not lie in the first quadrant

TO DETERMINE

The value of sinθ and tanθ

EVALUATION

Here it is given that

$\sf{ \cos \theta \: = t \: \: \: \: \: where \: \: 0 < t < 1}$

So cosθ is positive

Now cosθ is positive in first and fourth quadrant

Again it is also stated that θ does not lie in the first quadrant

So θ lies in Fourth Quadrant

So sinθ and tanθ both are negative

$\sf{ {\sin}^{2} \theta = 1 - { \cos}^{2} \theta}$

$\sf{ \implies \: {\sin}^{2} \theta = 1 - {t}^{2} }$

$\sf{ \implies \: {\sin}^{} \theta = - \sqrt{ 1 - {t}^{2}} }$

Again

$\displaystyle \: \sf{ \: {\tan}^{} \theta = \frac{ \sin \theta}{ \cos \theta} }$

$\displaystyle \: \sf{ \implies \: {\tan}^{} \theta = - \frac{ \sqrt{ 1 - {t}^{2}}}{t} }$

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