# CAT Practice : Trigonometry

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Max and Min value of the given expression

## Trigonometric Identities

Q.6:
In the above figure, the sheet of width W is folded along PQ such that R overlaps S Length of PQ can be written as :-

1. $\frac{w}{sin⁡\alpha (1+cos⁡2\alpha )}$
2. $\frac{w}{sin2⁡\alpha cos⁡\alpha }$
3. $\frac{w}{cos⁡\alpha (1+sin⁡2\alpha )}$
4. Any two of the above

Choice D. Any two of the above

## Detailed Solution

If you are quick at observing , this question can be solved just by looking at the options as ,
$\frac{w}{sin⁡∝(1+cos⁡2\alpha)}$ = $\frac{w}{sin⁡∝ (1+cos^2\alpha - sin^2\alpha)}$
= $\frac{w}{2cos^2\alpha} = \frac{w}{sin2⁡∝cos⁡∝}$ which is option (b)
Actually solving the question.
For R and S to overlap,
∆ PQR is congurent to ∆ PSQ
∴ ∠ QPR = ⁡∝ ; ∠ SQP = ∠ PQR = 90 - ⁡∝
=) ∠ RQT = 2⁡∝ (180°-(180°-2∝))
We can draw the figure as :-

∴In ∆ QRT
QT = QR cos2⁡∝
In ∆ PSQ,
SQ = PQ sin ⁡∝ = QR
∴ w = QT + SQ = PQ sin∝cos2∝ + PQ sin ∝
= $PQ sin \alpha (1 + cos 2\alpha)$
$\frac{w}{sin\alpha (1+cos⁡2\alpha)} = \frac{w}{sin2⁡∝cos⁡∝}$(as shown earlier)

Correct Answer: Any two of the above

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