Question
If
,then
Hint:
we have given trigonometry function. Which is
![fraction numerator sin cubed space theta minus cos cubed space theta over denominator sin space theta minus cos space theta end fraction minus fraction numerator cos begin display style space end style theta over denominator square root of 1 plus cot 2 space theta end root end fraction minus 2 tan space theta cot space theta equals negative 1](data:image/png;base64,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)
We have to find where θ is belongs. Solve the function and find the answer.
The correct answer is: ![theta element of open parentheses fraction numerator 3 pi over denominator 2 end fraction comma 2 pi close parentheses](data:image/png;base64,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)
Here, We have to find where θ is belongs.
Firstly, we have given
---(1)
[since,
]
We know that,
![a cubed minus b cubed equals left parenthesis a minus b right parenthesis left parenthesis a squared plus a b plus b squared right parenthesis](data:image/png;base64,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)
![1 plus fraction numerator sin theta cos theta minus cos theta over denominator open parentheses square root of left parenthesis 1 plus 2 cos squared theta – 1 end root close parentheses end fraction – 2 equals negative 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket c o t 2 theta equals 2 c o s squared theta – 1 right square bracket
1 plus fraction numerator sin theta cos theta minus cos theta over denominator open parentheses square root of 2 left parenthesis cos squared theta end root close parentheses end fraction – 2 equals negative 1
1 plus s i n theta c o s theta minus fraction numerator cos theta over denominator square root of 2 c o s theta – 2 end root end fraction equals negative 1
1 plus s i n theta c o s theta – fraction numerator 1 over denominator square root of 2 – 2 end root end fraction equals negative 1
s i n theta c o s theta – fraction numerator 1 over denominator square root of 2 end fraction equals 0
s i n theta c o s theta equals fraction numerator 1 over denominator square root of 2 end 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For every θ for sin and cos is
. so for But θ not possible in second quadrant because cos is negative,
Then θ always belongs to ( 0 ,
).
Therefore the correct answer is θ ∈ ( 0 ,
).
![fraction numerator sin cubed space theta minus cos cubed space theta over denominator sin space theta minus cos space theta end fraction minus fraction numerator cos begin display style space end style theta over denominator square root of 1 plus cot 2 space theta end root end fraction minus 2 tan space theta cot space theta equals negative 1](data:image/png;base64,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)
In this question, we have given equation, where we have to find where the θ belongs. For all value of sin θ cos θ = so θ always lies between ( 0 ,
) .
Related Questions to study
If ![6 cos space 2 theta plus 2 cos squared space theta divided by 2 plus 2 sin squared space theta equals 0 comma negative pi less than theta less than pi comma text then end text theta equals](data:image/png;base64,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)
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The most general solution of the equations is
The most general solution of the equations is
The general solution of the equation, is
The general solution of the equation, is
The most general solution of cot =
and cosec
= – 2 is :
In this question, we have given cotθ = -√3 and cosecθ = -2. Where θ for both is -π/6 and then write the general solution.
The most general solution of cot =
and cosec
= – 2 is :
In this question, we have given cotθ = -√3 and cosecθ = -2. Where θ for both is -π/6 and then write the general solution.
Let S =
The sun of all distinct solution of the equation ![square root of 3 sec space x plus cosec space x plus 2 left parenthesis tan space x minus cot space x right parenthesis equals 0](data:image/png;base64,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)
Let S =
The sun of all distinct solution of the equation ![square root of 3 sec space x plus cosec space x plus 2 left parenthesis tan space x minus cot space x right parenthesis equals 0](data:image/png;base64,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)
A wave motion has the function
The graph in figure shows how the displacement
at a fixed point varies with time
Which one of the labelled points shows a displacement equal to that at the position
at time ![t equals 0](data:image/png;base64,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)
![](data:image/png;base64,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)
A wave motion has the function
The graph in figure shows how the displacement
at a fixed point varies with time
Which one of the labelled points shows a displacement equal to that at the position
at time ![t equals 0](data:image/png;base64,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)
![](data:image/png;base64,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)
For
, the equation
has
For
, the equation
has
Which of the following pairs of compounds are functional isomers?
Which of the following pairs of compounds are functional isomers?
Let
be such that
and
Then
cannot satisfy
In this question we have to find the the which region ϕ cannot satisfy. In this question more than one option is correct. Here firstly solve the given equation. Remember that, , tan x + cot x = .
Let
be such that
and
Then
cannot satisfy
In this question we have to find the the which region ϕ cannot satisfy. In this question more than one option is correct. Here firstly solve the given equation. Remember that, , tan x + cot x = .
The number of all possible values of , where 0<
<
, which the system of equations
has a solution
with
, is
The number of all possible values of , where 0<
<
, which the system of equations
has a solution
with
, is
Roots of the equation are
Roots of the equation are
Isomers are possible for the molecular formula
Isomers are possible for the molecular formula
The number of solutions of the pair of equations in the interval [0, 2
] is
The number of solutions of the pair of equations in the interval [0, 2
] is