First slide

Trigonometric Identities

Question

Consider the system of equations $\log_{3} (\log_{2} x) + \log_{1 / 3} (\log_{1 / 2} y) = 1 and {xy}^{2} = 9$

Moderate

Question

The value of x lies in the interval

A

(200, 300)
B

(400, 500)
C

(700, 800)
D

none of these

Solution

$\begin{array}{l} \log_{3} (\log_{2} x) + \log_{1 / 3} (\log_{1 / 2} y) = 1 \\ \Rightarrow \log_{3} (\log_{2} x) - \log_{3} (- \log_{2} y) = 1 \\ \Rightarrow \log_{3} (- \frac{\log_{2} x}{\log_{2} y}) = 1 \\ \Rightarrow - \frac{\log_{2} x}{\log_{2} y} = 3 \\ \Rightarrow {xy}^{3} = 1 \\ Also, {xy}^{2} = 9 \\ \Rightarrow y = \frac{1}{9} \\ ∴ x = 729 \end{array}$

Question

The value of 1/y lies in the interval

A

(5, 7)
B

(7 , 10)
C

( 11, 15)
D

(25, 30)

Solution

$\begin{array}{l} \log_{3} (\log_{2} x) + \log_{1 / 3} (\log_{1 / 2} y) = 1 \\ \Rightarrow \log_{3} (\log_{2} x) - \log_{3} (- \log_{2} y) = 1 \\ \Rightarrow \log_{3} (- \frac{\log_{2} x}{\log_{2} y}) = 1 \\ \Rightarrow - \frac{\log_{2} x}{\log_{2} y} = 3 \\ \Rightarrow {xy}^{3} = 1 \\ Also, {xy}^{2} = 9 \\ \Rightarrow y = \frac{1}{9} \\ ∴ x = 729 \end{array}$

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