Let" "theta=arcsin(-5/13) θ=arcsin(−513)
This means that we are now looking for color(red)cottheta!cotθ!
=>sin(theta)=-5/13⇒sin(θ)=−513
Use the identity,
cos^2theta+sin^2theta=1cos2θ+sin2θ=1
**NB : ** sinthetasinθ is negative so thetaθ is also negative.
We shall the importance of this info later.
=>(cos^2theta+sin^2theta)/sin^2theta=1/sin^2theta⇒cos2θ+sin2θsin2θ=1sin2θ
=>cos^2theta/sin^2theta+1=1/sin^2theta⇒cos2θsin2θ+1=1sin2θ
=>cot^2theta+1=1/sin^2theta⇒cot2θ+1=1sin2θ
=>cot^2theta=1/sin^2x-1⇒cot2θ=1sin2x−1
=> cottheta=+-sqrt(1/sin^2(theta)-1)⇒cotθ=±√1sin2(θ)−1
=>cottheta=+-sqrt(1/(-5/13)^2-1)=+-sqrt(169/25-1)=+-sqrt(144/25)=+-12/5⇒cotθ=±
⎷1(−513)2−1=±√16925−1=±√14425=±125
WE saw the evidence previously that thetaθ should be negative only.
And since cotthetacotθ is odd =>cott(-A)=-cot(A)⇒cott(−A)=−cot(A) Where AA is a positive angle.
So, it becomes clear that cottheta=color(blue)+12/5cotθ=+125
REMEMBER what we called thetaθ was actually arcsin(-15/13)arcsin(−1513)
=>cot(arcsin(-5/13)) = color(blue)(12/5)⇒cot(arcsin(−513))=125
