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Question

he equation $sin invisible function application x open parentheses sin invisible function application x plus cos invisible function application x close parentheses equals k$ has real solutions if and only if $k$ is a real number such that

$0 \leq k \leq \frac{1 + \sqrt{2}}{2}$
$2 - \sqrt{3} \leq k \leq 2 + \sqrt{3}$
$0 \leq k \leq 2 - \sqrt{3}$
$\frac{1 - \sqrt{2}}{2} \leq k \leq \frac{1 + \sqrt{2}}{2}$

The correct answer is: $fraction numerator 1 minus square root of 2 over denominator 2 end fraction less or equal than k less or equal than fraction numerator 1 plus square root of 2 over denominator 2 end fraction$

$1 minus cos invisible function application 2 x plus sin invisible function application 2 x equals 2 k$
$rightwards double arrow sin invisible function application 2 x minus cos invisible function application 2 x equals 2 k minus 1$
$rightwards double arrow sin invisible function application open parentheses 2 x minus alpha close parentheses equals fraction numerator 2 k minus 1 over denominator square root of 2 end fraction$
$rightwards double arrow negative 1 less or equal than fraction numerator 2 k minus 1 over denominator square root of 2 end fraction less or equal than 1$
$rightwards double arrow fraction numerator 1 minus square root of 2 over denominator 2 end fraction less or equal than k less or equal than fraction numerator 1 plus square root of 2 over denominator 2 end fraction$

Related Questions to study

he equation $open parentheses cos invisible function application p minus 1 close parentheses x to the power of 2 end exponent plus open parentheses cos invisible function application p close parentheses x plus sin invisible function application p equals 0$ in the variable $x$ has real roots. Then $p$ can take any value in the interval

he equation $open parentheses cos invisible function application p minus 1 close parentheses x to the power of 2 end exponent plus open parentheses cos invisible function application p close parentheses x plus sin invisible function application p equals 0$ in the variable $x$ has real roots. Then $p$ can take any value in the interval

he value of expression $square root of 3 c o s e c blank 20 degree minus sec invisible function application 20 degree$ is equal to

he value of expression $square root of 3 c o s e c blank 20 degree minus sec invisible function application 20 degree$ is equal to

$f cos invisible function application x equals fraction numerator 2 cos invisible function application y minus 1 over denominator 2 minus cos invisible function application y end fraction comma blank w h e r e blank x comma blank y element of open parentheses 0 comma blank pi close parentheses comma blank t h e n tan invisible function application fraction numerator x over denominator 2 end fraction cot invisible function application fraction numerator y over denominator 2 end fraction blank i s blank e q u a l blank t o$

$f cos invisible function application x equals fraction numerator 2 cos invisible function application y minus 1 over denominator 2 minus cos invisible function application y end fraction comma blank w h e r e blank x comma blank y element of open parentheses 0 comma blank pi close parentheses comma blank t h e n tan invisible function application fraction numerator x over denominator 2 end fraction cot invisible function application fraction numerator y over denominator 2 end fraction blank i s blank e q u a l blank t o$

parallel

he value of $cos invisible function application fraction numerator pi over denominator 7 end fraction plus cos invisible function application fraction numerator 2 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 3 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 4 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 5 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 6 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 7 pi over denominator 7 end fraction$ is

he value of $cos invisible function application fraction numerator pi over denominator 7 end fraction plus cos invisible function application fraction numerator 2 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 3 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 4 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 5 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 6 pi over denominator 7 end fraction plus cos invisible function application fraction numerator 7 pi over denominator 7 end fraction$ is

f $alpha plus beta equals pi divided by 2$ and $beta plus gamma equals alpha comma$ then $tan invisible function application alpha blank e q u a l s$

f $alpha plus beta equals pi divided by 2$ and $beta plus gamma equals alpha comma$ then $tan invisible function application alpha blank e q u a l s$

he smallest $plus v e blank x$ satisfying the equation $log subscript cos invisible function application x end subscript invisible function application sin invisible function application x plus log subscript sin invisible function application x end subscript invisible function application cos invisible function application x equals 2$ is

he smallest $plus v e blank x$ satisfying the equation $log subscript cos invisible function application x end subscript invisible function application sin invisible function application x plus log subscript sin invisible function application x end subscript invisible function application cos invisible function application x equals 2$ is

parallel

f $log subscript 4 end subscript invisible function application 5 equals a blank a n d log subscript 5 end subscript invisible function application 6 equals b comma$ then $log subscript 3 end subscript invisible function application 2$ is equal to

f $log subscript 4 end subscript invisible function application 5 equals a blank a n d log subscript 5 end subscript invisible function application 6 equals b comma$ then $log subscript 3 end subscript invisible function application 2$ is equal to

f in $increment A B C comma blank 8 R to the power of 2 end exponent equals a to the power of 2 end exponent plus b to the power of 2 end exponent plus c to the power of 2 end exponent comma$ then the triangle $A B C$ is

f in $increment A B C comma blank 8 R to the power of 2 end exponent equals a to the power of 2 end exponent plus b to the power of 2 end exponent plus c to the power of 2 end exponent comma$ then the triangle $A B C$ is

The total number of solutions of $open vertical bar cot invisible function application x close vertical bar equals cot invisible function application x plus fraction numerator 1 over denominator sin invisible function application x end fraction comma blank x element of open square brackets 0 comma blank 3 pi close square brackets$ is equal to

The total number of solutions of $open vertical bar cot invisible function application x close vertical bar equals cot invisible function application x plus fraction numerator 1 over denominator sin invisible function application x end fraction comma blank x element of open square brackets 0 comma blank 3 pi close square brackets$ is equal to

parallel

If the inequality $sin to the power of 2 end exponent invisible function application x plus a cos invisible function application x plus a to the power of 2 end exponent greater than 1 plus cos invisible function application x$ holds for any $x element of R comma$ then the largest negative integral value of $a$ is

If the inequality $sin to the power of 2 end exponent invisible function application x plus a cos invisible function application x plus a to the power of 2 end exponent greater than 1 plus cos invisible function application x$ holds for any $x element of R comma$ then the largest negative integral value of $a$ is

$fraction numerator sin invisible function application 3 theta plus sin invisible function application 5 theta plus sin invisible function application 7 theta plus sin invisible function application 9 theta over denominator cos invisible function application 3 theta plus cos invisible function application 5 theta plus cos invisible function application 7 theta plus cos invisible function application 9 theta end fraction blank i s blank e q u a l blank t o$

$fraction numerator sin invisible function application 3 theta plus sin invisible function application 5 theta plus sin invisible function application 7 theta plus sin invisible function application 9 theta over denominator cos invisible function application 3 theta plus cos invisible function application 5 theta plus cos invisible function application 7 theta plus cos invisible function application 9 theta end fraction blank i s blank e q u a l blank t o$

If $a to the power of 4 end exponent blank. b to the power of 5 end exponent equals 1 comma$ then the value of $log subscript a end subscript invisible function application open parentheses a to the power of 5 end exponent b to the power of 4 end exponent close parentheses$ equals

If $a to the power of 4 end exponent blank. b to the power of 5 end exponent equals 1 comma$ then the value of $log subscript a end subscript invisible function application open parentheses a to the power of 5 end exponent b to the power of 4 end exponent close parentheses$ equals

parallel

If $open parentheses x plus 1 close parentheses to the power of log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses end exponent equals 100 open parentheses x plus 1 close parentheses comma$ then

If $open parentheses x plus 1 close parentheses to the power of log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses end exponent equals 100 open parentheses x plus 1 close parentheses comma$ then

The sum of all the solution of $cot invisible function application theta equals sin invisible function application 2 theta comma blank open parentheses theta not equal to n pi comma n blank i n t e g e r close parentheses comma blank 0 less or equal than theta less or equal than pi blank$ is

The sum of all the solution of $cot invisible function application theta equals sin invisible function application 2 theta comma blank open parentheses theta not equal to n pi comma n blank i n t e g e r close parentheses comma blank 0 less or equal than theta less or equal than pi blank$ is

The number of solutions of the equation $sin to the power of 3 end exponent invisible function application x cos invisible function application x plus sin to the power of 2 end exponent invisible function application x cos to the power of 2 end exponent invisible function application x plus sin invisible function application x cos to the power of 3 end exponent invisible function application x equals 1 comma blank$ in the interval $open square brackets 0 comma blank 2 pi close square brackets comma blank$ is

The number of solutions of the equation $sin to the power of 3 end exponent invisible function application x cos invisible function application x plus sin to the power of 2 end exponent invisible function application x cos to the power of 2 end exponent invisible function application x plus sin invisible function application x cos to the power of 3 end exponent invisible function application x equals 1 comma blank$ in the interval $open square brackets 0 comma blank 2 pi close square brackets comma blank$ is

parallel

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