Without solving, how do you determine the number of solutions for the equation $sectheta=(2sqrt3)/3$ in $-180^circ<=theta<=180^circ$ ?

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Without solving, how do you determine the number of solutions for the equation $sectheta=(2sqrt3)/3$ in $-180^circ<=theta<=180^circ$ ?

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Answer 1 · 2018-02-18T18:17:17

Two solutions.
See explanation.

Explanation:

$sec theta = 1/(cos theta)=((2sqrt3)/3)$

So you're really looking at where
$cos(theta)=(3)/(2sqrt3)=(sqrt3)/2$ (looks more familiar?)

$-180^@ = -pi$ radian
$180^@ = pi$ radian

Since the cosine function is "even", $cos (-x) = cos(x)$ for all $x$ .
So if there is a solution from 0 to $pi$ , there is also a solution from 0 to $-pi$ .

We can readily see that $0 < sqrt(3)/2 < 1$ ,
because $sqrt(3) < sqrt(4)$ .
We know that the range of the cosine function is between -1 and 1, so this is good news, this means that there is a solution (if the value was larger than 1, we would have had zero solution).

Therefore,
$cos(theta)=sqrt(3)/2$ has a solution that is between (0 and $180^@$ ), and another solution that is between (0 and $-180^@$ ).
Thus,
$sec theta= ((2sqrt3)/3)$ has two solutions in the range $-180^@ < theta < 180^@$ .
Q.E.D.

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Without solving, how do you determine the number of solutions for the equation $sectheta=(2sqrt3)/3$ in $-180^circ<=theta<=180^circ$ ?

1 Answer

Explanation:

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Without solving, how do you determine the number of solutions for the equation $sectheta=(2sqrt3)/3$ in $-180^circ<=theta<=180^circ$ ?