Answer
Verified
477.6k+ views
Hint- Here, we will be simplifying the LHS of the given equation by using the formulas which are \[\sin 2\theta = 2\left( {\sin \theta } \right)\left( {\cos \theta } \right)\] and \[\cos 2\theta = \left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right]\]. Then, with the help of the condition i.e., when \[\sin x = - 1\] then \[x = 2n\pi - \dfrac{\pi }{2}\] we will find the required values of $\theta $.
“Complete step-by-step answer:”
The given equation is \[{\left( {\sin \theta } \right)^3}\left( {\cos \theta } \right) - {\left( {\cos \theta } \right)^3}\left( {\sin \theta } \right) = \dfrac{1}{4}\].
This equation can be simplified as under
\[
\Rightarrow \left( {\sin \theta } \right)\left( {\cos \theta } \right)\left[ {{{\left( {\sin \theta } \right)}^2} - {{\left( {\cos \theta } \right)}^2}} \right] = \dfrac{1}{4} \\
\Rightarrow - 4\left( {\sin \theta } \right)\left( {\cos \theta } \right)\left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right] = 1 \\
\Rightarrow - 2\left[ {2\left( {\sin \theta } \right)\left( {\cos \theta } \right)} \right]\left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right] = 1{\text{ }} \to {\text{(1)}} \\
\]
As we know that \[\sin 2\theta = 2\left( {\sin \theta } \right)\left( {\cos \theta } \right)\] and \[\cos 2\theta = \left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right]\]
Using the above formulas, equation (1) becomes,
\[
\Rightarrow - 2\left[ {\sin 2\theta } \right]\left[ {\cos 2\theta } \right] = 1 \\
\Rightarrow - \left[ {2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right)} \right] = 1{\text{ }} \to {\text{(2)}} \\
\]
In the formula \[\sin 2\theta = 2\left( {\sin \theta } \right)\left( {\cos \theta } \right)\] replace \[\theta \] by \[2\theta \], we get
\[
\Rightarrow \sin 2\left( {2\theta } \right) = 2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right) \\
\Rightarrow \sin 4\theta = 2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right){\text{ }} \to {\text{(3)}} \\
\]
Using equation (3) in equation (2), we get
\[
\Rightarrow - \left[ {\sin 4\theta } \right] = 1 \\
\Rightarrow \sin 4\theta = - 1 \\
\]
Also we know that when \[\sin x = - 1\] then \[x = 2n\pi - \dfrac{\pi }{2}\] where $n \in Z$ (Z is set of integers)
Replacing x in the above condition by \[4\theta \], we get
\[
4\theta = 2n\pi - \dfrac{\pi }{2} \\
\Rightarrow \theta = \dfrac{{2n\pi }}{4} - \dfrac{\pi }{{2 \times 4}} \\
\Rightarrow \theta = \dfrac{{n\pi }}{2} - \dfrac{\pi }{8}{\text{ where }}n \in Z \\
\]
Note- In these types of problems, we convert the trigonometric functions of smaller angle (i.e., \[\theta \] in this case) in the given equation to the trigonometric functions of larger angle (i.e. \[4\theta \] in this case). Also, in this problem \[\sin 4\theta = - 1\] is coming after simplification which means the values of \[4\theta \] which are possible are \[2n\pi - \dfrac{\pi }{2}\] where n belongs to the set of integers.
“Complete step-by-step answer:”
The given equation is \[{\left( {\sin \theta } \right)^3}\left( {\cos \theta } \right) - {\left( {\cos \theta } \right)^3}\left( {\sin \theta } \right) = \dfrac{1}{4}\].
This equation can be simplified as under
\[
\Rightarrow \left( {\sin \theta } \right)\left( {\cos \theta } \right)\left[ {{{\left( {\sin \theta } \right)}^2} - {{\left( {\cos \theta } \right)}^2}} \right] = \dfrac{1}{4} \\
\Rightarrow - 4\left( {\sin \theta } \right)\left( {\cos \theta } \right)\left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right] = 1 \\
\Rightarrow - 2\left[ {2\left( {\sin \theta } \right)\left( {\cos \theta } \right)} \right]\left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right] = 1{\text{ }} \to {\text{(1)}} \\
\]
As we know that \[\sin 2\theta = 2\left( {\sin \theta } \right)\left( {\cos \theta } \right)\] and \[\cos 2\theta = \left[ {{{\left( {\cos \theta } \right)}^2} - {{\left( {\sin \theta } \right)}^2}} \right]\]
Using the above formulas, equation (1) becomes,
\[
\Rightarrow - 2\left[ {\sin 2\theta } \right]\left[ {\cos 2\theta } \right] = 1 \\
\Rightarrow - \left[ {2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right)} \right] = 1{\text{ }} \to {\text{(2)}} \\
\]
In the formula \[\sin 2\theta = 2\left( {\sin \theta } \right)\left( {\cos \theta } \right)\] replace \[\theta \] by \[2\theta \], we get
\[
\Rightarrow \sin 2\left( {2\theta } \right) = 2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right) \\
\Rightarrow \sin 4\theta = 2\left( {\sin 2\theta } \right)\left( {\cos 2\theta } \right){\text{ }} \to {\text{(3)}} \\
\]
Using equation (3) in equation (2), we get
\[
\Rightarrow - \left[ {\sin 4\theta } \right] = 1 \\
\Rightarrow \sin 4\theta = - 1 \\
\]
Also we know that when \[\sin x = - 1\] then \[x = 2n\pi - \dfrac{\pi }{2}\] where $n \in Z$ (Z is set of integers)
Replacing x in the above condition by \[4\theta \], we get
\[
4\theta = 2n\pi - \dfrac{\pi }{2} \\
\Rightarrow \theta = \dfrac{{2n\pi }}{4} - \dfrac{\pi }{{2 \times 4}} \\
\Rightarrow \theta = \dfrac{{n\pi }}{2} - \dfrac{\pi }{8}{\text{ where }}n \in Z \\
\]
Note- In these types of problems, we convert the trigonometric functions of smaller angle (i.e., \[\theta \] in this case) in the given equation to the trigonometric functions of larger angle (i.e. \[4\theta \] in this case). Also, in this problem \[\sin 4\theta = - 1\] is coming after simplification which means the values of \[4\theta \] which are possible are \[2n\pi - \dfrac{\pi }{2}\] where n belongs to the set of integers.
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
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
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
Why is there a time difference of about 5 hours between class 10 social science CBSE