Ravi Kant Kumar

Evaluate cos 47° cos 13° − sin 47° sin 13° Question Evaluate: \[ \cos 47^\circ \cos 13^\circ-\sin 47^\circ \sin 13^\circ \] Solution Using the identity: \[ \cos A \cos B-\sin A \sin B=\cos(A+B) \] Here, \[ A=47^\circ,\qquad B=13^\circ \] Therefore, \[ \cos 47^\circ \cos 13^\circ-\sin 47^\circ \sin 13^\circ \] \[ =\cos(47^\circ+13^\circ) \] \[ =\cos 60^\circ

Evaluate sin 78° cos 18° − cos 78° sin 18° Question Evaluate: \[ \sin 78^\circ \cos 18^\circ-\cos 78^\circ \sin 18^\circ \] Solution Using the identity: \[ \sin A \cos B-\cos A \sin B=\sin(A-B) \] Here, \[ A=78^\circ,\qquad B=18^\circ \] Therefore, \[ \sin 78^\circ \cos 18^\circ-\cos 78^\circ \sin 18^\circ \] \[ =\sin(78^\circ-18^\circ) \] \[ =\sin 60^\circ

If sin A = 1/2 and cos B = √3/2, Find tan(A+B) and tan(A−B) Question If \[ \sin A=\frac{1}{2} \] and \[ \cos B=\frac{\sqrt{3}}{2} \] where \[ \frac{\pi}{2} < A < \pi \] and \[ 0 < B < \frac{\pi}{2} \] find the following: (i) \(\tan(A+B)\) (ii) \(\tan(A-B)\) Solution Given: \[ \sin A=\frac{1}{2} \] Using

If sin A = 1/2 and cos B = 12/13, Find tan(A−B) Question If \[ \sin A=\frac{1}{2} \] and \[ \cos B=\frac{12}{13} \] where \[ \frac{\pi}{2} < A < \pi \] and \[ \frac{3\pi}{2} < B < 2\pi \] find: \[ \tan(A-B) \] Solution Given: \[ \sin A=\frac{1}{2} \] Using \[ \sin^2 A+\cos^2 A=1 \]

If tan A = 3/4 and cos B = 9/41, Find tan(A+B) Question If \[ \tan A=\frac{3}{4} \] and \[ \cos B=\frac{9}{41} \] where \[ \pi < A < \frac{3\pi}{2} \] and \[ 0 < B < \frac{\pi}{2} \] find: \[ \tan(A+B) \] Solution Given: \[ \tan A=\frac{3}{4} \] Since \[ \pi < A <

If cos A = -24/25 and cos B = 3/5, Find sin(A+B) and cos(A+B) Question If \[ \cos A=-\frac{24}{25} \] and \[ \cos B=\frac{3}{5} \] where \[ \pi < A < \frac{3\pi}{2} \] and \[ \frac{3\pi}{2} < B < 2\pi \] find: (i) \(\sin(A+B)\) (ii) \(\cos(A+B)\) Solution Given: \[ \cos A=-\frac{24}{25} \] Using \[ \sin^2

If sin A = 12/13 and sin B = 4/5, Find sin(A+B) and cos(A+B) If sin A = 12/13 and sin B = 4/5, Find sin(A+B) and cos(A+B) Question If \[ \sin A=\frac{12}{13} \] and \[ \sin B=\frac{4}{5} \] where \[ \frac{\pi}{2} < A < \pi \] and \[ 0 < B < \frac{\pi}{2} \]

If sin A = 4/5 and cos B = 5/13, Find sin(A+B), cos(A+B), sin(A−B), cos(A−B) If sin A = 4/5 and cos B = 5/13, Find sin(A+B), cos(A+B), sin(A−B), cos(A−B) Question If \[ \sin A = \frac{4}{5} \] and \[ \cos B = \frac{5}{13} \] where \[ 0 < A, B < \frac{\pi}{2} \] find

Sketch the Graph of f(x) = tan 2x Question: Sketch the graph of the following function : \[ f(x)=\tan2x \] Solution: We know that \[ \tan\theta=\frac{\sin\theta}{\cos\theta} \] Therefore \[ f(x)=\tan2x \] The tangent graph increases from \(-\infty\) to \(+\infty\) between consecutive asymptotes. Whenever \[ \cos2x=0 \] the function becomes undefined. Thus vertical asymptotes occur at

Sketch the Graph of f(x) = cosec²x Question: Sketch the graph of the following function : \[ f(x)=\cosec^2x \] Solution: We know that \[ \cosec^2x=\frac{1}{\sin^2x} \] Since square of cosecant is always positive, the graph always lies above the x-axis. Whenever \[ \sin x=0 \] the function becomes undefined. Thus vertical asymptotes occur at \[