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Find the acute angle \[\theta \] through which coordinate axes should be rotated for the point \[A(2,4)\] to attend new abscissa \[4\].
A). \[\tan \theta = \dfrac{3}{4}\]
B). \[\tan \theta = \dfrac{5}{6}\]
C). \[\tan \theta = \dfrac{7}{8}\]
D). None of these

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Last updated date: 16th Apr 2024
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MVSAT 2024
Answer
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Hint: We need to find the acute angle \[\theta \] through which coordinate axes should be rotated for the point \[A(2,4)\] to attend new abscissa \[4\]. To find the new position we need to use the identity given below in the hint section.
Formula used:
The acute angle\[\theta \] through which coordinate axes should be rotated for the point \[A(x,y)\] to attend new abscissa \[X\] is given by,
\[X = x\cos \theta + y\sin \theta \]

Complete step-by-step solution:
We have to find the acute angle \[\theta \] through which coordinate axes should be rotated for the point \[A(2,4)\] to attend new abscissa \[4\].
Let us note the given information,
\[A(x,y) = A(2,4)\]
Let us use the below identity,
The acute angle \[\theta \] through which coordinate axes should be rotated for the point \[A(x,y)\] to attend new abscissa \[X\] is given by,
\[X = x\cos \theta + y\sin \theta \]
On putting values \[(x,y) = (2,4)\] in above equation we get,
\[4 = 2\cos \theta + 4\sin \theta \]
On dividing the equation by $2$ on both sides we get,
\[2 = \cos \theta + 2\sin \theta \]
On rearranging the terms on both sides of the equation we get,
\[\cos \theta = 2 - 2\sin \theta \]
On squaring both side we get,
\[{\cos ^2}\theta = {\left( {2 - 2\sin \theta } \right)^2}\]
On performing square of the bracket using the formula \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\] on R.H.S. we get,
\[{\cos ^2}\theta = 4 - 8\sin \theta + 4{\sin ^2}\theta \]
On putting value \[{\cos ^2}\theta = 1 - {\sin ^2}\theta \] in above equation we get,
\[1 - {\sin ^2}\theta = 4 - 8\sin \theta + 4{\sin ^2}\theta \]
On arranging all the terms on L.H.S. and performing the addition we get,
\[5{\sin ^2}\theta - 8\sin \theta + 3 = 0\]
On splitting the middle term to factorize the equation we get,
\[5{\sin ^2}\theta - 5\sin \theta - 3\sin \theta + 3 = 0\]
On taking common terms out we get,
\[5\sin \theta (1 - \sin \theta ) - 3(1 - \sin \theta ) = 0\]
On taking common term out we get,
\[\left( {\sin \theta - 1} \right)\left( {5\sin \theta - 3} \right) = 0\]
On equating both factors to zero we get,
\[\sin \theta - 1 = 0,5\sin \theta - 3 = 0\]
On rearranging the terms of the equation we get,
\[\sin \theta = 1,\sin \theta = \dfrac{3}{5}\]
On considering first value,
\[\sin \theta = 1\]
From above value we can write that,
\[\tan \theta = \infty \]
Thus this value \[\sin \theta = 1\]is invalid.
On considering second value,
\[\sin \theta = \dfrac{3}{5}\]
From above value we can write that,
\[\cos \theta = \dfrac{4}{5}\]
On taking ratios of both values we can write that,
\[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
On putting both values we can write that,
\[\tan \theta = \dfrac{{\dfrac{3}{5}}}{{\dfrac{4}{5}}}\]
On cancelling the common denominator and performing the operation we get,
\[\tan \theta = \dfrac{3}{4}\]
Hence option A)$\tan \theta = \dfrac{3}{4}$ is correct.

Note: We need to calculate the new angle using the identity and by performing operations. We need to choose a valid value and we need to discard the value which gives the answer which is out of the range.