Question

If tan θ= 12/5 then find the value of 1+sinθ /1-sinθ​

Answer 1

Pythagorean Theorem

We can directly find any trigonometric values if we bring a right triangle, since $5^2 + 12^2 = 13^2$. (A 5-12-13 Pythagorean triangle.)

 

Giving:-

• $\sin \theta = \dfrac{12}{13}$
• $\cos \theta = \dfrac{5}{13}$
• $\tan \theta = \dfrac{12}{5}$

 

But $\theta$ is not always in the 1st Quadrant, the angle above is just acute ones.

 

Case i) $\theta$ lies on the 1st Quadrant.

For the $\sin \theta = \dfrac{12}{13}$ case, this gives $\dfrac{1 + \sin \theta}{1 - \sin \theta} = 25$.

 

Case ii) $\theta$ lies on the 3rd Quadrant.

For the $\sin \theta = - \dfrac{12}{13}$ case, this gives $\dfrac{1 + \sin \theta}{1 - \sin \theta} = -27$.

 

Calculation Process

Let's break down a rational polynomial $\dfrac{1+t}{1-t}$.

$\implies \dfrac{1 + t}{1 - t} = - 1 + \dfrac{2}{1 - t}$

 

Note that:-

Actually, we could solve this with $\dfrac{1 + \sin \theta}{1 - \sin \theta} = (\sec \theta - \tan \theta)^2$, but we proved that it is possible to solve without it. (Shortcut Process)

It is better to keep away from formulas if you don't want mistakes to be made.

Answer 2

I hope it helps all.

#Hard work