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Question

f $a comma blank b blank$ and $blank A$ are given in a triangle and $c subscript 1 end subscript comma blank c subscript 2 end subscript$ are the possible values of the third side, then $c subscript 1 end subscript superscript 2 end superscript plus c subscript 2 end subscript superscript 2 end superscript minus 2 c subscript 1 end subscript c subscript 2 end subscript cos invisible function application A equals$ is equal to

$4 a^{2} \sin 2 A$
$4 a^{2} \sin^{2} A$
$4 a^{2} \cos 2 A$
$4 a^{2} \cos^{2} A$

The correct answer is: $4 a to the power of 2 end exponent cos to the power of 2 end exponent invisible function application A$

We have $blank cos invisible function application A equals fraction numerator b to the power of 2 end exponent plus c to the power of 2 end exponent minus a to the power of 2 end exponent over denominator 2 b c end fraction$
$rightwards double arrow c to the power of 2 end exponent minus 2 blank b c cos invisible function application A plus b to the power of 2 end exponent minus a to the power of 2 end exponent equals 0 comma blank$ which is quadratic in $blank ´ c ´$
$therefore c subscript 1 end subscript plus c subscript 2 end subscript equals 2 b cos invisible function application A blank$ and $blank c subscript 1 end subscript c subscript 2 end subscript equals b to the power of 2 end exponent minus a to the power of 2 end exponent$ (i)
$therefore c subscript 1 end subscript superscript 2 end superscript plus c subscript 2 end subscript superscript 2 end superscript minus 2 c subscript 1 end subscript c subscript 2 end subscript cos invisible function application 2 A$
$equals open parentheses c subscript 1 end subscript plus c subscript 2 end subscript close parentheses to the power of 2 end exponent minus 2 c subscript 1 end subscript c subscript 2 end subscript minus 2 c subscript 1 end subscript c subscript 2 end subscript cos invisible function application 2 A$ [using Eq.(i)]
$equals open parentheses c subscript 1 end subscript plus c subscript 2 end subscript close parentheses to the power of 2 end exponent minus 2 c subscript 1 end subscript c subscript 2 end subscript open parentheses 1 plus cos invisible function application 2 A close parentheses$
$equals 4 b to the power of 2 end exponent cos to the power of 2 end exponent invisible function application A minus 2 open parentheses b to the power of 2 end exponent minus a to the power of 2 end exponent close parentheses. blank 2 cos to the power of 2 end exponent invisible function application A equals 4 a to the power of 2 end exponent cos to the power of 2 end exponent invisible function application A$

Related Questions to study

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

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

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

parallel

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

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

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 $tan invisible function application alpha blank e q u a l s$

parallel

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

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

parallel

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

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

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

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

parallel

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

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

parallel

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