Answer
Verified
378.9k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE