Educational

Find the Maximum and Minimum Value of 12 cos x + 5 sin x + 4 Find the Maximum and Minimum Value of 12 cos x + 5 sin x + 4 Question: Find the maximum and minimum values of the following trigonometrical expression: \[ 12\cos x + 5\sin x + 4 \] Solution We […]

Find the Maximum and Minimum Value of 12 sin x − 5 cos x Find the Maximum and Minimum Value of 12 sin x − 5 cos x Question: Find the maximum and minimum values of the following trigonometrical expression: \[ 12\sin x – 5\cos x \] Solution We use the standard result: \[ a\sin

If α and β are solutions of a tanx + b secx = c, Find sin(α+β) and cos(α+β) Question If \[ a\tan x+b\sec x=c \] and \[ \alpha,\beta \] are two solutions of the equation, find: \[ \sin(\alpha+\beta) \quad \text{and} \quad \cos(\alpha+\beta) \] Solution Given, \[ a\tan x+b\sec x=c \] \[ a\frac{\sin x}{\cos x} +

If tanx = (sinα − cosα)/(sinα + cosα), Show that sinα + cosα = √2 cosx Question If \[ \tan x = \frac{\sin\alpha-\cos\alpha} {\sin\alpha+\cos\alpha} \] show that: \[ \sin\alpha+\cos\alpha = \sqrt2\cos x \] Proof Given, \[ \tan x = \frac{\sin\alpha-\cos\alpha} {\sin\alpha+\cos\alpha} \] Using \[ \sin\alpha-\cos\alpha = \sqrt2\sin\left(\alpha-\frac{\pi}{4}\right) \] and \[ \sin\alpha+\cos\alpha = \sqrt2\cos\left(\alpha-\frac{\pi}{4}\right) \] \[

If tan of one part is λ times the other, Show that sinθ = ((λ+1)/(λ−1)) sinϕ Question If angle \[ \theta \] is divided into two parts such that the tangent of one part is \[ \lambda \] times the tangent of the other, and \[ \phi \] is their difference, show that: \[ \sin\theta

If tanα = x+1 and tanβ = x−1, Show that 2cot(α−β) = x² Question If \[ \tan\alpha=x+1 \] and \[ \tan\beta=x-1 \] show that: \[ 2\cot(\alpha-\beta)=x^2 \] Proof Using \[ \tan(\alpha-\beta) = \frac{\tan\alpha-\tan\beta} {1+\tan\alpha\tan\beta} \] \[ = \frac{(x+1)-(x-1)} {1+(x+1)(x-1)} \] \[ = \frac{2} {1+x^2-1} \] \[ = \frac{2}{x^2} \] Therefore, \[ \cot(\alpha-\beta) = \frac{x^2}{2} \]

If sinα sinβ − cosα cosβ + 1 = 0, Prove that 1 + cotα tanβ = 0 Question If \[ \sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0 \] prove that: \[ 1+\cot\alpha\tan\beta=0 \] Proof Given, \[ \sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0 \] \[ \sin\alpha\sin\beta-\cos\alpha\cos\beta=-1 \] Using \[ \cos(\alpha+\beta) = \cos\alpha\cos\beta-\sin\alpha\sin\beta \] \[ -\cos(\alpha+\beta)=-1 \] \[ \cos(\alpha+\beta)=1 \] \[ \alpha+\beta=0 \] Therefore, \[ \tan(\alpha+\beta)=0 \]

Prove that: 1/[cos(x−a)cos(x−b)] = [tan(x−b) − tan(x−a)]/sin(a−b) Question Prove that: \[ \frac{1}{\cos(x-a)\cos(x-b)} = \frac{\tan(x-b)-\tan(x-a)} {\sin(a-b)} \] Proof R.H.S. \[ = \frac{\tan(x-b)-\tan(x-a)} {\sin(a-b)} \] \[ = \frac{ \frac{\sin(x-b)}{\cos(x-b)} – \frac{\sin(x-a)}{\cos(x-a)} } {\sin(a-b)} \] \[ = \frac{ \sin(x-b)\cos(x-a) – \cos(x-b)\sin(x-a) } {\sin(a-b)\cos(x-a)\cos(x-b)} \] Using \[ \sin C\cos D-\cos C\sin D = \sin(C-D) \] \[ = \frac{ \sin[(x-b)-(x-a)]

Prove that: 1/[sin(x−a)cos(x−b)] = [cot(x−a) + tan(x−b)]/cos(a−b) Question Prove that: \[ \frac{1}{\sin(x-a)\cos(x-b)} = \frac{\cot(x-a)+\tan(x-b)} {\cos(a-b)} \] Proof R.H.S. \[ = \frac{\cot(x-a)+\tan(x-b)} {\cos(a-b)} \] \[ = \frac{ \frac{\cos(x-a)}{\sin(x-a)} + \frac{\sin(x-b)}{\cos(x-b)} } {\cos(a-b)} \] \[ = \frac{ \cos(x-a)\cos(x-b) + \sin(x-a)\sin(x-b) } {\cos(a-b)\sin(x-a)\cos(x-b)} \] Using \[ \cos C\cos D+\sin C\sin D = \cos(C-D) \] \[ = \frac{ \cos[(x-a)-(x-b)]

Prove that: 1/[sin(x−a)sin(x−b)] = [cot(x−a) − cot(x−b)]/sin(a−b) Question Prove that: \[ \frac{1}{\sin(x-a)\sin(x-b)} = \frac{\cot(x-a)-\cot(x-b)} {\sin(a-b)} \] Proof R.H.S. \[ = \frac{\cot(x-a)-\cot(x-b)} {\sin(a-b)} \] \[ = \frac{ \frac{\cos(x-a)}{\sin(x-a)} – \frac{\cos(x-b)}{\sin(x-b)} } {\sin(a-b)} \] \[ = \frac{ \cos(x-a)\sin(x-b) – \sin(x-a)\cos(x-b) } {\sin(a-b)\sin(x-a)\sin(x-b)} \] Using \[ \sin C\cos D-\cos C\sin D = \sin(C-D) \] \[ = \frac{ \sin[(x-b)-(x-a)]