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Find the number of solutions for \cos x=\frac{x}{5} 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

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Simultaneous Trigonometric Equations -

Simultaneous Trigonometric Equations

 

We can divide the problems related to Simultaneous Trigonometric Equations into two categories:

  1. If two equations satisfies simultaneously having only one unknown.

  2. If two equations satisfies simultaneously having two unknowns.

Let us go through some illustrations to understand how to solve Simultaneous Trigonometric Equations

Illustrations 1

The most general value of x which satisfies the equation cos x = -½ and cot x = 1/√3.

First, find the value of x which satisfies two given equations separately

cos x = -½ ⇒ x = 2π/3 and  4π/3 [in 0 to 2π]

cot x =  1/√3 ⇒ x = π/3 and  4π/3 [in 0 to 2π]

Now select the value x which satisfies both the equation

Here common value is 4π/3,  

Hence, the required solution is x = 2nπ + 4π/3.

Illustrations 2

If two equations given r sin x = 4 and r = 3 sin x - 1, this equation satisfies if  -π ≤ x ≤ π , then the possible solution of the pair (r, x) is

 Here two equation given   r sin x = 4 and r = 3 sin x - 1

Eliminating x from above equation, we get

\\\mathrm{r=3\cdot \frac{4}{r}-1}\\\mathrm{\Rightarrow r^2=12-r}\\\mathrm{\Rightarrow r=3,-4}

Now,

\\\mathrm{r\sin x=4\Rightarrow \sin x = \frac{4}{-4}=-1}\\\mathrm{and,\sin x = \frac{4}{3}\;which \;is \;not\;possible}\\\mathrm{solve\;\sin x =-1}\\\mathrm{\Rightarrow x=-\frac{\pi }{2}}

Hence the required pair is (-4, -π/2 )

-

\cos x=\frac{x}{5}\\ -1 \leq \cos x \leq 1\\ -5 \leq x \leq 5\\ \text{ at x=} 2\pi\\ \frac{x}{5}>1

 

Graph - 

 

By the graph we can say the number of solutions for this equation is 3.

 

 

Posted by

Pankaj Sanodiya

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