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If \[\sin {15^ \circ } = \cos \left( {n \times {{15}^ \circ }} \right)\], then n=______.
A. 1
B. 2
C. 5
D. 0

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Last updated date: 13th Jun 2024
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Answer
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Hint: When two lines one real axis and the other imaginary axis passes perpendicular through a circle, the circle is divided into four quadrants. Into Quadrant I where both x and y axis are positive, Quadrant II here x axis is negative and y axis is positive, Quadrant III here both the x axis and y axis are negative and in Quadrant IV x-axis is positive and y-axis is negative.
In case of the trigonometric functions in Quadrant I all the functions are positive, in Quadrant II Sin and Cosec functions are positive and other functions are negative, in Quadrant III tan and cot functions are positive and other are negative and in the case of Quadrant IV Cos and Sec functions are positive and other being negative.
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In this question for the given equation we will cofunction identities of the trigonometric function.

Complete step by step solution: \[\sin {15^ \circ } = \cos \left( {n \times {{15}^ \circ }} \right)\]
By the rule of cofunction identity,\[\sin \theta = \cos \left( {{{90}^ \circ } - \theta } \right)\]
We can write as:
\[
  \sin \theta = \cos \left( {{{90}^ \circ } - \theta } \right) \\
  \sin {15^ \circ } = \cos \left( {{{90}^ \circ } - {{15}^ \circ }} \right) \\
   = \cos \left( {{{75}^ \circ }} \right) \\
   \\
 \]
We can also write, \[\cos \left( {{{75}^ \circ }} \right) = \cos \left( {5 \times {{15}^ \circ }} \right) = \cos \left( {n \times {{15}^ \circ }} \right)\]
Hence, the value of n=5,
Where \[\sin {15^ \circ }\]and \[\cos \left( {{{75}^ \circ }} \right)\]both lie in the Quadrant I where all the trigonometric functions are positive. Quadrant I lie in the range of \[0 - {90^ \circ }\].

Note: The co-function identities show the relationship between the sin, cos, tan, cosine, sec and cot function. The value of the trigonometric function for an angle is equal to the value of the co-function of the complement.