1. Tardigrade
  2. Question
  3. Mathematics
  4. If a sin θ-b cos θ=0 and a sin 3 θ+b cos 3 θ= sin 2 θ, then find the value of a2+b2.

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $a \sin \theta-b \cos \theta=0$ and $a \sin ^3 \theta+b \cos ^3 \theta=\sin 2 \theta$, then find the value of $a^2+b^2$.

Trigonometry

Solution:

Given, $a \sin \theta-b \cos \theta=0$
$a \sin \theta=b \cos \theta$
$a \sin ^3 \theta+b \cos ^3 \theta=\sin 2 \theta$
$a \sin \theta \sin ^2 \theta+b \cos \theta \cos ^2 \theta=2 \sin \theta \cos \theta$
$b \cos \theta \sin ^2 \theta+b \cos \theta \cos ^2 \theta=2 \sin \theta \cos \theta$
$b \cos \theta\left(\sin ^2 \theta+\cos ^2 \theta\right)=2 \sin \theta \cos \theta$
$b \cos \theta=2 \sin \theta \cos \theta$
$b=2 \sin \theta$
$\text { (1) } \Rightarrow a \sin \theta=2 \sin \theta \cos \theta$
$ a=2 \cos \theta $
$ a^2+b^2 =(2 \cos \theta)^2+(2 \sin \theta)^2 $
$ =4 \cos ^2 \theta+4 \sin ^2 \theta $
$ =4(\cos ^2 \theta+\sin ^2 \theta)=4$