Maths-
General
Easy

Question

Let P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text  and  end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket be two sets. Then

  1. P subset of Q text  and  end text P not equal to straight empty set
  2. Q ⊂⊂ P
  3. P subset of Q
  4. P=Q

Hint:

In the two sets theta is derived from the given trigonometric equations in the respective sets. So, we will first simplify the equation then will get the relation between them.

The correct answer is: P=Q


    Given two sets P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text  and  end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket
    F r o m space P comma space sin theta space minus cos theta equals square root of 2 cos theta 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 space space space space space F r o m space Q comma space sin theta plus cos theta equals square root of 2 sin theta
rightwards double arrow sin theta equals square root of 2 cos theta plus cos theta 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 space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals square root of 2 sin theta minus sin theta
rightwards double arrow sin theta equals cos theta open parentheses square root of 2 plus 1 close parentheses 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 space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals sin theta open parentheses square root of 2 minus 1 close parentheses
rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals square root of 2 plus 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 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 space rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals fraction numerator 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction
rightwards double arrow tan theta equals square root of 2 plus 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 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 space space space space rightwards double arrow tan theta equals fraction numerator square root of 2 plus 1 over denominator 2 minus 1 end fraction equals square root of 2 plus 1
H e n c e comma space P equals Q

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    A long cylindrical tube carries a highly polished piston and has a side opening. A tuning fork of frequency n is sounded at the sound heard by the listener charges if the piston is moves in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of n is

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    When piston moves a distancex subscript 1 end subscript, path difference change by 2 xs.
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    cot space fraction numerator 7 pi over denominator 16 end fraction plus 2 cot space fraction numerator 3 pi over denominator 8 end fraction plus cot space fraction numerator 15 pi over denominator 16 end fraction
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equals open parentheses fraction numerator sin squared begin display style straight pi over 16 end style minus cos squared begin display style straight pi over 16 end style over denominator cos begin display style straight pi over 16 end style sin begin display style straight pi over 16 end style end fraction close parentheses plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction
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    If square root of 2 cos space A equals cos space B plus cos cubed space B text  and  end text square root of 2 sin text end text A equals sin space B minus sin cubed space B then sin (A-B) =

    GIVEN-
    ​   √2 cos cos cos3 B..... equation 1
    ∴      cos = 1/√2 (cos cos3 B) ........ (Dividing both sides by √2)

    √2 sin sin B - sin3 B..... equation 2

    ∴      sin = 1/√2 (sin B - sin3 B) ........ (Dividing both sides by √2)
    TO FIND-
    Sin (A - B)
    SOLUTION-
    We know that-
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    ∴ Sin (A - B) = [1/√2 (sin B - sin3 B) * cos B] - [1/√2 (cos cos3 B) * sin B] ...... (From Equations i & ii)
    ∴ Sin (A - B) = [1/√2 * (sin B Cos B - sin3 B Cos B)] - [1/√2 * (sin B cos cos3 B * sin B)]
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - (1/√2 sin B cos + 1/√2 cos3 B * sin B)
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos - 1/√2 cos3 B * sin B
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos B - 1/√2 cos3 B * sin B
    ∴ Sin (A - B) = -1/√2 sin3 B Cos B - 1/√2 cos3 B * sin B

    ∴ Sin (A - B) = -1/√2 sin B Cos B (sin2 B +cos2 B)
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rightwards double arrow 2 open parentheses cos squared A plus sin squared A close parentheses equals cos squared B plus sin squared B plus open parentheses cos squared B plus sin squared B close parentheses cubed minus 3 open parentheses cos squared B sin squared B close parentheses open parentheses cos squared B plus sin squared B close parentheses plus 2 open parentheses cos to the power of 4 B minus sin to the power of 4 B close parentheses
rightwards double arrow 2 equals 1 plus 1 minus 3 over 4 sin squared 2 B plus 2 cos 2 B space space space space open square brackets 4 cos squared B sin squared B equals sin squared 2 B comma cos to the power of 4 B minus sin to the power of 4 B equals cos 2 B space close square brackets space space space open square brackets cos squared B plus sin squared B equals 1 close square brackets
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rightwards double arrow 0 equals negative 3 open parentheses 1 minus cos squared 2 B close parentheses plus 8 cos 2 B space space space space open square brackets cos squared 2 B plus sin squared 2 B equals 1 close square brackets
rightwards double arrow 0 equals 3 cos squared 2 B minus 3 plus 8 cos 2 B
rightwards double arrow 3 cos squared 2 B plus 8 cos 2 B minus 3 equals 0
rightwards double arrow 3 cos squared 2 B plus 9 cos 2 B minus cos 2 B minus 3 equals 0
rightwards double arrow 3 cos 2 B open parentheses cos 2 B plus 3 close parentheses minus 1 open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow open parentheses 3 cos 2 B minus 1 close parentheses open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow cos 2 B equals 1 third space o r space cos 2 B equals negative 3 open parentheses n o t space p o s s i b l e close parentheses
N o w comma space w e space k n o w space t h a t space open parentheses cos squared 2 B plus sin squared 2 B close parentheses equals 1
S o comma space 1 over 9 plus sin squared 2 B equals 1 rightwards double arrow sin squared 2 B equals 1 minus 1 over 9 equals 8 over 9
sin 2 B equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction
t h e r e f o r e comma space sin open parentheses A minus B close parentheses equals fraction numerator negative sin 2 B over denominator 2 square root of 2 end fraction equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction cross times fraction numerator 1 over denominator 2 square root of 2 end fraction equals plus-or-minus 1 third

    If square root of 2 cos space A equals cos space B plus cos cubed space B text  and  end text square root of 2 sin text end text A equals sin space B minus sin cubed space B then sin (A-B) =

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    GIVEN-
    ​   √2 cos cos cos3 B..... equation 1
    ∴      cos = 1/√2 (cos cos3 B) ........ (Dividing both sides by √2)

    √2 sin sin B - sin3 B..... equation 2

    ∴      sin = 1/√2 (sin B - sin3 B) ........ (Dividing both sides by √2)
    TO FIND-
    Sin (A - B)
    SOLUTION-
    We know that-
    Sin (A - B) = (Sin A * Cos B) - (Cos A * Sin B)
    ∴ Sin (A - B) = [1/√2 (sin B - sin3 B) * cos B] - [1/√2 (cos cos3 B) * sin B] ...... (From Equations i & ii)
    ∴ Sin (A - B) = [1/√2 * (sin B Cos B - sin3 B Cos B)] - [1/√2 * (sin B cos cos3 B * sin B)]
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - (1/√2 sin B cos + 1/√2 cos3 B * sin B)
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos - 1/√2 cos3 B * sin B
    ∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos B - 1/√2 cos3 B * sin B
    ∴ Sin (A - B) = -1/√2 sin3 B Cos B - 1/√2 cos3 B * sin B

    ∴ Sin (A - B) = -1/√2 sin B Cos B (sin2 B +cos2 B)
                        = -1/2√2 sin2B
    Now squaring and adding equation 1 and 2 we get
    2 cos squared A plus 2 sin squared A equals cos squared B plus cos to the power of 6 B plus 2 cos to the power of 4 B plus sin squared B plus sin to the power of 6 B minus 2 sin to the power of 4 B
rightwards double arrow 2 open parentheses cos squared A plus sin squared A close parentheses equals cos squared B plus sin squared B plus open parentheses cos squared B plus sin squared B close parentheses cubed minus 3 open parentheses cos squared B sin squared B close parentheses open parentheses cos squared B plus sin squared B close parentheses plus 2 open parentheses cos to the power of 4 B minus sin to the power of 4 B close parentheses
rightwards double arrow 2 equals 1 plus 1 minus 3 over 4 sin squared 2 B plus 2 cos 2 B space space space space open square brackets 4 cos squared B sin squared B equals sin squared 2 B comma cos to the power of 4 B minus sin to the power of 4 B equals cos 2 B space close square brackets space space space open square brackets cos squared B plus sin squared B equals 1 close square brackets
rightwards double arrow 0 equals negative 3 sin squared 2 B plus 8 cos 2 B
rightwards double arrow 0 equals negative 3 open parentheses 1 minus cos squared 2 B close parentheses plus 8 cos 2 B space space space space open square brackets cos squared 2 B plus sin squared 2 B equals 1 close square brackets
rightwards double arrow 0 equals 3 cos squared 2 B minus 3 plus 8 cos 2 B
rightwards double arrow 3 cos squared 2 B plus 8 cos 2 B minus 3 equals 0
rightwards double arrow 3 cos squared 2 B plus 9 cos 2 B minus cos 2 B minus 3 equals 0
rightwards double arrow 3 cos 2 B open parentheses cos 2 B plus 3 close parentheses minus 1 open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow open parentheses 3 cos 2 B minus 1 close parentheses open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow cos 2 B equals 1 third space o r space cos 2 B equals negative 3 open parentheses n o t space p o s s i b l e close parentheses
N o w comma space w e space k n o w space t h a t space open parentheses cos squared 2 B plus sin squared 2 B close parentheses equals 1
S o comma space 1 over 9 plus sin squared 2 B equals 1 rightwards double arrow sin squared 2 B equals 1 minus 1 over 9 equals 8 over 9
sin 2 B equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction
t h e r e f o r e comma space sin open parentheses A minus B close parentheses equals fraction numerator negative sin 2 B over denominator 2 square root of 2 end fraction equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction cross times fraction numerator 1 over denominator 2 square root of 2 end fraction equals plus-or-minus 1 third
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