University Maths Solution

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Maths Question
Question 1

$\text{If si}{{\text{n}}^{-1}}x=2{{\cos }^{-1}}y\text{ prove that }{{x}^{2}}=4y(1-{{y}^{2}})$

Question 2

$\text{Show that si}{{\text{n}}^{-1}}\left( \frac{24}{25} \right)+2{{\cos }^{-1}}\left( \frac{3}{5} \right)=\pi $

Question 3

$\text{Prove without using table }{{\sin }^{-1}}\left( \frac{3}{5} \right)-{{\cos }^{-1}}\left( \frac{63}{65} \right)=2{{\tan }^{-1}}\left( \frac{1}{5} \right)$

Question 4

$\text{If 2}{{\sin }^{-1}}x={{\tan }^{-1}}y\text{ prove that }y(1-2{{x}^{2}})=2x\sqrt{1-{{x}^{2}}}$

Question 5

$\begin{align}  & \text{If }A,B,C\text{ are angles of a triangle show that } \\ & sin2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C \\\end{align}$

Question 6

$\begin{align}  & \text{If }A,B,C\text{ are angles of a triangle show that } \\ & \sin A+\sin B+\sin C=4\cos \tfrac{A}{2}\cos \tfrac{B}{2}\cos \tfrac{C}{2} \\\end{align}$

Question 7

$\begin{align}  & \text{If }A,B,C\text{ are angles of a triangle show that } \\ & 3\cos \tfrac{A}{2}\cos \tfrac{B}{2}\cos \tfrac{C}{2}-\cos \tfrac{3A}{2}\cos \tfrac{3B}{2}\cos \tfrac{3C}{2}=\left( {{\sin }^{3}}A+{{\sin }^{3}}A+{{\sin }^{3}}C \right) \\\end{align}$ 

Question 8

$\begin{align}  & \text{If }\alpha \text{ and }\beta \text{ are two unequal angles which satisfy the } \\ & \text{equation }a\cos \alpha +b\sin \alpha =c,\text{ show that} \\ & a)\sin \frac{\alpha +\beta }{2}\sec \frac{\alpha -\beta }{2}=\frac{b}{c} \\ & b)\tan \frac{\alpha }{2}\sec \frac{\beta }{2}=\frac{c-a}{c+a} \\\end{align}$

Question 9

$\text{Where possible solve the  triangle }c=2.718,\text{ }b=3.142,\text{ }C={{56}^{\circ }}.18'$

Question 10

$\text{ Where possible solve the  triangle }c=4.13,\text{ }b=5.62,\text{ }C={{61}^{\circ }}23'$

Question 11

$\text{Where possible solve the  triangle }a=651,\text{ }c=792,\text{ }C={{73}^{\circ }}.22'$

Question 12

$\text{Using the sine rule formula, prove that }\frac{b-c}{a}=\frac{\sin (\tfrac{B-C}{2})}{\cos \tfrac{A}{2}}$

Question 13

$\begin{align}  & \text{Prove that }4\cos \theta \cos 3\theta +1=\frac{\sin 5\theta }{\sin \theta }\text{, Hence or otherwise, find all} \\ & \text{ the values of }\theta \text{ in the range 0}\le \theta \le \text{18}{{\text{0}}^{\circ }}\text{ for which }\cos \theta \cos 3\theta =\frac{1}{2} \\\end{align}$

Question 14

$\text{If }A,\text{ }B\,\text{and }C\text{ are angles of a triangle prove that }\cot (A+B)+\cot C=0$

Question 15

$\begin{align}  & \text{Given that }5\cos \theta +12\sin \theta =R\cos (\theta -\alpha ),\text{ where }R\text{ and }\alpha \text{ are independent } \\ & \text{of }\theta \text{ and }R\text{ is positive, obtain the value of }R\text{ and Hence or otherwise, find the} \\ & \text{value of }\theta \text{ between }-{{180}^{\circ }}\,\text{and }{{180}^{\circ }}\text{ which satisfy the equation } \\ & 5\cos \theta +12\sin \theta =3\tfrac{1}{4} \\\end{align}$

Question 16

$\begin{align}  & \text{Prove that } \\ & \text{(a) }\sin \theta =\frac{2{{\tan }^{2}}\tfrac{\theta }{2}}{1+{{\tan }^{2}}\tfrac{\theta }{2}} \\ & (b)\text{ }\cos \theta =\frac{1-{{\tan }^{2}}\tfrac{\theta }{2}}{1+{{\tan }^{2}}\tfrac{\theta }{2}} \\ & \text{Hence or otherwise find the value of }\theta \text{ and} \\ & \text{2}\pi \text{ which satsify the equation }55\cos \theta +48\sin \theta =51 \\\end{align}$

Question 17

$\begin{align}  & \text{Express }\sin x-\sqrt{3}\cos x\,\text{in the form }R\sin (x-\alpha ) \\ & \text{Hence solve }\sqrt{3}\sin x-3\cos x=3 \\\end{align}$

Question 18

$\begin{align}  & \text{Find all the values of }x\text{ between }0\text{ and }2\pi \text{ radian} \\ & \text{which satisfy the equation }(2+\sqrt{3})\sin 3x-\cos 3x=1 \\\end{align}$