Question
If
,then
Hint:
In this question, we have given trigonometry function. Which is
and belongs to [ 0, 2 π]. We have to find where θ is belongs. Solve the function and find the answer.
The correct answer is: ![theta element of open parentheses pi over 2 comma pi close parentheses minus open curly brackets fraction numerator 3 pi over denominator 4 end fraction close curly brackets](data:image/png;base64,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)
Here , we have to find where θ lies in this equation.
Firstly, we have given
![fraction numerator left parenthesis sin cubed theta minus cos cubed theta right parenthesis over denominator left parenthesis s i n theta minus c o s theta right parenthesis end fraction minus fraction numerator cos theta over denominator square root of left parenthesis 1 plus c o t squared theta right parenthesis end root end fraction minus 2 t a n theta c o t theta equals negative 1.......... left parenthesis 1 right parenthesis
fraction numerator left parenthesis sin cubed theta minus cos cubed theta right parenthesis over denominator left parenthesis s i n theta minus c o s theta right parenthesis end fraction minus fraction numerator cos theta over denominator left parenthesis 1 plus c o t squared theta right parenthesis end fraction minus 2 equals negative 1 space left square bracket s i n c e comma t a n theta equals fraction numerator 1 over denominator c o t theta end fraction right square bracket](data:image/png;base64,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)
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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)
![fraction numerator left parenthesis sin cubed theta minus cos cubed theta right parenthesis over denominator left parenthesis s i n theta minus c o s theta right parenthesis end fraction equals left parenthesis s i n theta minus c o s theta right parenthesis fraction numerator left parenthesis sin squared theta plus sin theta cos theta plus cos squared theta right parenthesis over denominator left parenthesis s i n theta minus c o s theta right parenthesis end fraction](data:image/png;base64,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)
= 1 + 2 sin θ cos θ
So, we can write, in eq (1)
![fraction numerator 1 space plus space space sin space theta space cos space theta space minus space cos space theta over denominator fraction numerator left parenthesis sin squared theta plus cos to the power of theta right parenthesis over denominator sin squared theta end fraction 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 open square brackets c o t squared theta equals fraction numerator cos squared theta over denominator sin squared theta end fraction close square brackets](data:image/png;base64,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)
sin θ cos θ - |sin θ| cos θ = 0
cos θ ( sin θ - | sin θ| ) = 0
hence, sin θ = | sin θ |
which is always true for sinx ≥ 0 otherwise it is true for x=0,
,
,2π
Therefore, since we also need x≠
,
for tanx and x≠0,2π for cotx all the solutions are given by x∈(0,π) with x≠
and x≠
.
The correct answer is θ ϵ (0 ,π)- {
,
}
In this question, we have to find the where is θ lies. Here, always true for sinx ≥ 0 otherwise it is true for x=0,,
,2π and since we also need x≠
,
for tanx and x≠0,2π for cotx all the solutions.
Related Questions to study
Number of solutions of the equation in the interval [0, 2
] is :
Number of solutions of the equation in the interval [0, 2
] is :
If
,then
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 ,
) .
If
,then
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 ,
) .
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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)
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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)
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