Question

if sin theta +cos theta =k then 2(sin^3 theta+cos^3 theta )=
a) 3k-4k^3
b)2k+4k^3
c)3k+k^3
d)3k-k^3​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

If sin θ + cos θ = k then 2 ( sin³ θ + cos³ θ ) =

a) 3k - 4k³

b) 2k + 4k³

c) 3k + k³

d) 3k - k³

EVALUATION

Here it is given that

sin θ + cos θ = k

Now Squaring both sides we get

sin² θ + cos² θ + 2 sin θ cos θ = k²

$\sf{ \implies \: 1 + 2 \sin \theta \cos \theta = {k}^{2} }$

$\sf{ \implies \: 2 \sin \theta \cos \theta = {k}^{2} - 1}$

Now

2 ( sin³ θ + cos³ θ )

$\sf{ = 2( \sin \theta + \cos \theta)( {\sin}^{2} \theta + {\cos}^{2} \theta - \sin \theta \cos \theta ) }$

$\sf{ = 2( \sin \theta + \cos \theta)( 1 - \sin \theta \cos \theta ) }$

$\sf{ = ( \sin \theta + \cos \theta)( 2 - 2\sin \theta \cos \theta ) }$

$\sf{ = k \times ( 2 - {k}^{2} + 1) }$

$\sf{ = k \times ( 3 - {k}^{2} ) }$

$\sf{ = 3k - {k}^{3} }$

FINAL ANSWER

Hence the correct option is d) 3k - k³

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