If left| cos theta left{ sin theta + sqrt{sin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $u = \cos 	heta \left\{ \sin 	heta + \sqrt{\sin^2 	heta + \sin^2 \alpha} ight\}$.Rearranging the terms,we get $u - \sin 	heta \cos 	het

Vedclass · Accepted Answer

$\sqrt{1 + \sin^2 \alpha}$

Vedclass · Answer

(B) Let $u = \cos 	heta \left\{ \sin 	heta + \sqrt{\sin^2 	heta + \sin^2 \alpha} ight\}$.Rearranging the terms,we get $u - \sin 	heta \cos 	heta = \cos 	heta \sqrt{\sin^2 	heta + \sin^2 \alpha}$.Squaring both sides:$(u - \sin 	heta \cos 	heta)^2 = \cos^2 	heta (\sin^2 	heta + \sin^2 \alpha)$.$u^2 - 2u \sin 	heta \cos 	heta + \sin^2 	heta \cos^2 	heta = \sin^2 	heta \cos^2 	heta + \cos^2 	heta \sin^2 \alpha$.$u^2 - 2u \sin 	heta \cos 	heta = \cos^2 	heta \sin^2 \alpha$.Dividing by $\cos^2 	heta$ (assuming $\cos 	heta 
eq 0$):$u^2 \sec^2 	heta - 2u 	an 	heta = \sin^2 \alpha$.Using $\sec^2 	heta = 1 + 	an^2 	heta$:$u^2(1 + 	an^2 	heta) - 2u 	an 	heta - \sin^2 \alpha = 0$.$u^2 	an^2 	heta - 2u 	an 	heta + (u^2 - \sin^2 \alpha) = 0$.Since $	an 	heta$ is real,the discriminant $D \ge 0$:$(-2u)^2 - 4(u^2)(u^2 - \sin^2 \alpha) \ge 0$.$4u^2 - 4u^4 + 4u^2 \sin^2 \alpha \ge 0$.Dividing by $4u^2$ (assuming $u 
eq 0$):$1 - u^2 + \sin^2 \alpha \ge 0$.$u^2 \le 1 + \sin^2 \alpha$.Therefore,$|u| \le \sqrt{1 + \sin^2 \alpha}$.Thus,$k = \sqrt{1 + \sin^2 \alpha}$.

If $\left| \cos \theta \left\{ \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right\} \right| \le k$,then the value of $k$ is

Similar Questions

$cosec^2 \theta = \frac{4xy}{(x + y)^2}$ is true if and only if

If $0 \leq a, b \leq 3$ and the equation $x^2+4+3 \cos (ax+b)=2x$ has a real solution,then the value of $(a+b)$ is

If $P+Q+R=\frac{\pi}{4}$,then $\cos \left(\frac{\pi}{8}-P\right)+\cos \left(\frac{\pi}{8}-Q\right)+\cos \left(\frac{\pi}{8}-R\right)=$

If $A + B + C = \pi$ $(A, B, C > 0)$ and the angle $C$ is obtuse,then:

The minimum value of $3 \cos x + 4 \sin x + 5$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module