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Separate into real and imaginary parts of $ sin^{-1} e^{i\theta}$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Prove $tan \Bigg[ ilog \bigg ( \dfrac{a-ib}{a+ib} \bigg) \Bigg] \;=\; \dfrac{2ab}{a^2-b^2} $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $ cosec \bigg( \dfrac{\pi}{4} + ix \bigg) \;=\; u + iv $. Prove $ (u^2+v^2)^2 \;=\; 2(u^2-v^2) $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $cos(x+iy)cos(u+iv)=1 $ where x,y,u,v are real, then show that $ tan^2y \; cosh^2v \;=\; sin^2 u $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Separate into real & imaginary parts: $ tan^{-1} (e^{i\theta}) $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If tan $ \bigg( \dfrac{\pi}{4} + iv \bigg) \;=\; re^{i\theta}$ Show : $ (i) r=1 \; (ii) \; tan \theta = sinh2v \; (iii) tanh v = tan \dfrac{\theta}{2} $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Complex Numbers : Solve $x^7+x^4+i(x^3+1)=0.$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Solve the equation $x^6+1=0.$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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($x+1 )^6 + (x-1)^6\;=\; 0 $ are given by $ -icot\Bigg[ \dfrac{(2k+1) \pi}{12} \Bigg] \; \; \;, K=0,1,2,3,5 $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $\tan (x+iy) = a+ib$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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show that $\log[e^{i\alpha}+e^{i\beta}] =\log[2\cos(\dfrac {\alpha-\beta}2) + i(\dfrac {\alpha+\beta}2)]$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Show that $\tan^{-1}(\dfrac {x+iy}{x-iy})=\dfrac {\pi}4 +\dfrac i2 \log (\dfrac {x+y}{x-y})$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $\log\tan x=y$ then prove that $\sin h(n+1)y + \sin h(n-1)y=2\sin h\space\space ny\space \space cosec 2x $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Solve the equation $7\cos hx + 8\sin hx=1$ for real values of x
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Prove that $\cos^6\theta + \sin^6\theta =\dfrac 18 (3\cos4\theta+5) $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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IF $\alpha,\beta$ are the roots of the equation $x^2-2\sqrt3 x+4=0$ find the value of $\alpha^3+\beta^3$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Considering only the principal value, if $ (1+itan \alpha )^{1+itan\beta} $ is real, prove that its value is $ (sec \alpha)^{sec^2 \beta} $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $ tan \Big[ log(x+iy) \Big] \;=\; a+ib , $ prove that $ tan[log(x^2+y^2)] \;=\; \dfrac{2a}{1-a^2-b^2} \; \; \; where \; a^2+b^2 \ne 1 $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $ \alpha=1+i \;,\; \beta=1-i \; and \; cot \theta \;=\; x+1 \;,\; $ Prove that $ (x+\alpha)^n + (x+\beta)^n\;=\; (\alpha+\beta) cosn\theta \; cosec^n\theta $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Show that for real values of a and b, $ e^{2aicot^{-1} b } \Bigg[ \dfrac{bi-1}{bi+1} \Bigg]^{-a} \;=\; 1 $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $i^{i^{i^{\ldots \infty }}} \;=\; A+iB $ , then then prove that $tan( \dfrac{\pi A}{2} )= \dfrac{B}{2}\; and \; A^2+B^2=e^{-\pi B} $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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If $ y=(x-1)^n $ then prove that $ y+ \dfrac{y_1}{1!} + \dfrac{y_2}{2!} + \dfrac{y_3}{3!}+\ldots+\dfrac{y_n}{n!} \; = \; x^n $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Expansion Standard Series : Prove that $log(sec x) = \ \dfrac{x^2}{2} + \dfrac{x^4}{12} + \dfrac{x^6}{45}+\ldots $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Expand $ \dfrac{x}{e^x-1} \; in \; powers \; of \; x. \\ Hence \; Prove \; \dfrac{x}{2} \bigg[ \dfrac{e^x+1}{e^x-1} \bigg] \; = \; 1 + \dfrac{1}{12}x^2-\dfrac{1}{720}x^4+\ldots $
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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Expansion Standard Series : Use Taylor$'$s Theorem and arrange the equation in powers of $ x \; : \; 7 + (x+2) + 3(x+2)^3 + (x+2)^4 - (x+2)^5$
engineering mathematics updated 5.7 years ago by teamques10 ★ 70k
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