Question

If 13cos theta -5=0 find sin theta +cos theta / sin theta -cos theta​

Answer 1

Explanation:

Need to FinD :

• We have to find the value of (sinθ + cosθ)/(sinθ - cosθ), when 13 cosθ - 5 = 0.

`\red{\frak{Given}} \begin{cases} & \sf {13\ cos \theta\ -\ 5\ =\ 0\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \big\lgroup Can\ also\ be\ written\ as \big\rgroup} \\ & \sf {cos \theta\ =\ {\footnotesize{(5)/(13)}}} \end{cases}`

Here, we're asked to find out the value of (sinθ + cosθ)/(sinθ - cosθ), when 13 cosθ - 5 = 0. In order to find the solution we're gonna use trigonometric ratios to find the value of sinθ and cosθ. Let us consider, a right angled triangle, say PQR.

Where,

• PQ = Opposite side
• QR = Adjacent side
• RP = Hypotenuse
• ∠Q = 90°
• ∠C = θ

As we know that, 13 cosθ - 5 = 0 which is stated in the question. So, it can also be written as cosθ = 5/13. As per the cosine ratio, we know that,

`\rightarrow {\underline{\boxed{\red{\sf{cos \theta\ =\ (Adjacent\ side)/(Hypotenuse)}}}}}`

Since, we know that,

• cosθ = 5/13
• QR (Adjacent side) = 5
• RP (Hypotenuse) = 13

So, we will find the PQ (Opposite side) in order to estimate the value of sinθ. So, by using the Pythagoras Theorem, we will find the PQ.

Therefore,

`\red \bigstar {\underline{\underline{\pmb{\sf{According\ to\ Question:-}}}}}`

`\rule{200}{3}`

`\sf \dashrightarrow {(PQ)^2\ +\ (QR)^2\ =\ (RP)^2} \\ \\ \\ \sf \dashrightarrow {(PQ)^2\ +\ (5)^2\ =\ (13)^2} \\ \\ \\ \sf \dashrightarrow {(PQ)^2\ +\ 25\ =\ 169} \\ \\ \\ \sf \dashrightarrow {(PQ)^2\ =\ 169\ -\ 25} \\ \\ \\ \sf \dashrightarrow {(PQ)^2\ =\ 144} \\ \\ \\ \sf \dashrightarrow {PQ\ =\ √(144)} \\ \\ \\ \dashrightarrow {\underbrace{\boxed{\pink{\frak{PQ\ (Opposite\ side)\ =\ 12}}}}_{\sf \blue{\tiny{Required\ value}}}}`

∴ Hence, the value of PQ (Opposite side) is 12. Now, in order to determine it's value, we will use the sine ratio.

`\rightarrow {\underline{\boxed{\red{\sf{sin \theta\ =\ (Opposite\ side)/(Hypotenuse)}}}}}`

Where,

• Opposite side = 12
• Hypotenuse = 13

Therefore,

`\sf \rightarrow {sin \theta\ =\ (12)/(13)}`

Now, we have the values of sinθ and cosθ, that are 12/13 and 5/13 respectively. Now, finally we will find out the value of the following.

`\rightarrow {\underline{\boxed{\red{\sf{(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)}}}}}`

• By substituting the values, we get,

`\rule{200}{3}`

`\sf \dashrightarrow {(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)\ =\ {\footnotesize{(\Big( (12)/(13)\ +\ (5)/(13) \Big))/(\Big( (12)/(13)\ -\ (5)/(13) \Big))}}} \\ \\ \\ \sf \dashrightarrow {(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)\ =\ {\footnotesize{((17)/(13))/((7)/(13))}}} \\ \\ \\ \sf \dashrightarrow {(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)\ =\ (17)/(13) * (13)/(7)} \\ \\ \\ \sf \dashrightarrow {(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)\ =\ \frac{17}{\cancel{13}} * \frac{\cancel{13}}{7}} \\ \\ \\ \dashrightarrow {\underbrace{\boxed{\pink{\frak{(sin \theta\ +\ cos \theta)/(sin \theta\ -\ cos \theta)\ =\ (17)/(7)}}}}_{\sf \blue{\tiny{Required\ value}}}}`

∴ Hence, the required answer is 17/7.