$\mathbf{a}$ and $\mathbf{b}$ are unit vectors such that $\mathbf{a}+2 \mathbf{b}$ is also a unit vector. If $ heta$ is the angle between a and $\mathbf{b}$, then $\sin heta+\cos ^3 heta+ an ^5 heta$ is equal to

Question

$\mathbf{a}$ and $\mathbf{b}$ are unit vectors such that $\mathbf{a}+2 \mathbf{b}$ is also a unit vector. If $	heta$ is the angle between a and $\mathbf{b}$, then $\sin 	heta+\cos ^3 	heta+	an ^5 	heta$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Given, }|\mathbf{a}|=|\mathbf{b}|=|\mathbf{a}+2 \mathbf{b}|=1 \ & \qquad \begin{aligned}|\mathbf{a}+2 \mathbf{b}|^2 & =|\mathbf{a}|^2+4\left|\mathbf{b}^2ight|+4|\mathbf{a}||\mathbf{b}| \cos 	heta \ 1 & =1+4+4 \cos 	heta \ \cos 	heta & =-1 \Rightarrow 	heta=\pi \ \sin 	heta & +\cos ^3 	heta+	an ^3 	heta \ & =\sin \pi+\cos ^3 \pi+	an ^3 \pi \ =0 & +(-1)^3+0=-1\end{aligned}\end{aligned}$

$\mathbf{a}$ and $\mathbf{b}$ are unit vectors such that $\mathbf{a}+2 \mathbf{b}$ is also a unit vector. If…

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