if sec theta+ tan theta =p then find the value of sin theta in terms of p.
Answers
Answer:
p²-1
Step-by-step explanation:
sec theta +tan theta =p_______ 1
sec theta - tan theta=1/p______2
1-2
2 tan theta=p²-1/p
tan theta =p²-1/2p
tan theta =sin theta/cos theta
comparing on both sides
sin theta=p²-1
EXPLANATION.
⇒ secθ + tanθ = p. - - - - - (1).
As we know that,
Formula of :
⇒ 1 + tan²θ = sec²θ.
⇒ sec²θ - tan²θ = 1.
⇒ (x² - y²) = (x - y)(x + y).
Using this formula in the equation, we get.
⇒ (secθ - tanθ)(secθ + tanθ) = 1.
Put the value in the equation, we get.
⇒ (secθ - tanθ)(p) = 1.
⇒ secθ - tanθ = 1/p. - - - - - (2).
From equation (1) & (2), we get.
⇒ secθ + tanθ = p. - - - - - (1).
⇒ secθ - tanθ = 1/p. - - - - - (2).
Adding both the equation, we get.
⇒ 2secθ = p + 1/p.
⇒ 2secθ = (p² + 1)/p.
⇒ secθ = (p² + 1)/2p.
⇒ secθ = 1/cosθ.
⇒ cosθ = (2p)/(p² + 1).
As we know that,
⇒ cosθ = Base/Hypotenuse = (2p)/(p² + 1).
By using Pythagoras theorem, we get.
⇒ H² = (Perpendicular)² + B².
⇒ [(p² + 1)]² = (Perpendicular)² + (2p)².
⇒ [p⁴ + 1 + 2p²] = (Perpendicular)² + 4p².
⇒ p⁴ + 1 + 2p² - 4p² = (Perpendicular)².
⇒ p⁴ - 2p² + 1 = (Perpendicular)².
⇒ (1 - p²)² = (Perpendicular)².
⇒ (Perpendicular) = (1 - p²).
⇒ sinθ = Perpendicular/Hypotenuse.
⇒ sinθ = (1 - p²)/(1 + p²).
MORE INFORMATION.
(1) = sin²θ + cos²θ = 1.
(2) = 1 + tan²θ = sec²θ.
(3) = 1 + cot²θ = cosec²θ.