Answer
351.9k+ 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
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)