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  • The number of distinct solutions of the equation, <img alt="\log _{\frac{1}{2}}\left | \sin x \right |=2-\log _{\frac{1}{2}}\left | \cos x \right |" src="https://entrancecorner.oncodecogs.com/

The number of distinct solutions of the equation, \log _{\frac{1}{2}}\left | \sin x \right |=2-\log _{\frac{1}{2}}\left | \cos x \right | in the interval \left [ 0,2\pi \right ], is _________.

 

Option: 1

4


Option: 2

2


Option: 3

10


Option: 4

8


Answers (1)

best_answer

 

 

Logarithmic Inequalities -

Logarithmic inequalities:

\mathrm{\log_ax>\log_ay=\left\{\begin{matrix} \mathrm{x>y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x<y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

\mathrm{\log_ax>y=\left\{\begin{matrix} \mathrm{x>a^y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x<a^y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

\mathrm{\log_ax<y=\left\{\begin{matrix} \mathrm{x<a^y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x>a^y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

Generally, from these inequalities, we can conclude that logarithmic functions are monotonically increasing for a >1 and decreasing for 0 < a < 1. Basic property which we must remember about log is that argument ( means x or y) must be positive.

 

Double Angle Formula and Reduction Formula -

Double Angle Formula and Reduction Formula

\begin{aligned} \sin (2 \theta) &=2 \sin \theta \cos \theta \\&=\frac{2\tan\theta}{1+\tan^2\theta} \\\cos (2 \theta) &=\cos ^{2} \theta-\sin ^{2} \theta \\ &=1-2 \sin ^{2} \theta \\ &=2 \cos ^{2} \theta-1\\&=\frac{1-\tan^2\theta}{1+\tan^2\theta} \\ \tan (2 \theta) &=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \end{aligned}

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Trigonometric Equations -

Trigonometric Equations

Trigonometric equations are, as the name implies, equations that involve trigonometric functions.

Solution of Trigonometric Equation

The value of an unknown angle which satisfies the given trigonometric equation is called a solution or root of the equation. For example, 2sin? = 1, clearly ? = 300 satisfies the equation; therefore, 300 is a solution of the equation. Now trigonometric equation ususally has infinite solutions due to periodic nature of trigonometric functions. So this equation also has (360+30)o,(720+30)o,(-360+30)o and so on, as its solutions.

Principal Solution

The solutions of a trigonometric equation that lie in the interval [0, 2π). For example, 2sin? = 1 , then the two values of sin? between 0 and 2π are  π/6 and 5π/6. Thus, π/6 and 5π/6 are the principal solutions of equation 2sin? = 1.

General Solution

As trigonometric functions are periodic, solutions are repeated within each period, so, trigonometric equations may have an infinite number of solutions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

Some Important General Solutions of Equations

\mathbf{Equations}

\mathbf{Solution}

\sin\theta=0

\theta=n \pi, \quad n \in \mathbb{I}

\cos\theta=0

\theta=(2 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}

\tan\theta=0

\theta=n \pi, \quad n \in \mathbb{I}

\sin\theta=1

\theta=(4 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}

\cos\theta=1

\theta=2 n \pi, \quad n \in \mathbb{I}

\sin\theta=-1

{\theta}=(4 n-1) \frac{\pi}{2}, \quad n \in \mathbb{I}

\cos\theta=-1

\theta=(2 n+1) \pi, \quad n \in \mathbb{I}

\cot\theta=0

\theta=(2 n+1) \frac{\pi}{2}, \quad n \in \mathbb{I}

-

 

\\\log _{\frac{1}{2}}\left(\left|\sin \left(x\right)\right|\right)+\log _{\frac{1}{2}}\left(\left|\cos \left(x\right)\right|\right)=2\\\log _{\frac{1}{2}}\left(\left|\sin \left(x\right)\right|\left|\cos \:\left(x\right)\right|\right)=2\\\log _{\frac{1}{2}}\left(\left|\sin \left(x\right)\right|\left|\cos \left(x\right)\right|\right)=\log \:_{\frac{1}{2}}\left(\frac{1}{4}\right)\\|\sin x||\cos x|=\frac{1}{4}\\\sin2\theta=\frac{1}{2}\\x=\frac{\pi }{12}+\pi n,\:x=\frac{5\pi }{12}+\pi n

Total number of solution is 8

 

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