Question

If sec theta + tan theta =p find the value of sec theta, tan theta, sin theta in terms of p

Answer 1

Answer:

secβ = p²+1/2p

tanβ = p²-1/2p

sinβ = p²-1/p²+1

Step-by-step explanation:

Given: secβ + tanβ = p

⇒ $\frac{1}{cos\beta} + \frac{sin\beta}{cos\beta}$ = p

or $\frac{1+sin\beta}{cos\beta}$ = p

or $\frac{1+sin\beta}{\sqrt{1-sin^{2}\beta}}$ = p

or $\frac{1+sin\beta}{\sqrt{(1-sin\beta)(1+sin\beta)}}$ = p

or $\sqrt{\frac{1+sin\beta}{1-sin\beta}}$ = p

or $\frac{1+sin\beta}{1-sin\beta}$ = p²

or 1+sinβ = p²(1-sinβ) = p²-p²sinβ

or sinβ+p²sinβ = p²-1

or sinβ(1+p²) = p²-1

⇒ sinβ = p²-1/p²+1 = Perpendicular/hypotenuse

⇒ Perpendicular = p²-1 ; Hypotenuse = p²+1 ; Base = 2p

⇒ tanβ = Perpendicular/Base

or tanβ = p²-1/2p

And, secβ = Hypotenuse/Base

⇒ secβ = p²+1/2p

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