Show that
$\text{\hspace{0.17em}}\frac{\mathrm{cot}\text{\hspace{0.17em}}\theta}{\mathrm{csc}\text{\hspace{0.17em}}\theta}=\mathrm{cos}\text{\hspace{0.17em}}\theta .$
Create an identity for the expression
$\text{\hspace{0.17em}}2\mathrm{tan}\text{\hspace{0.17em}}\theta \mathrm{sec}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ by rewriting strictly in terms of sine.
There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression:
Using algebra to simplify trigonometric expressions
We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.
For example, the equation
$\text{\hspace{0.17em}}\left(\mathrm{sin}\text{\hspace{0.17em}}x+1\right)\left(\mathrm{sin}\text{\hspace{0.17em}}x-1\right)=0\text{\hspace{0.17em}}$ resembles the equation
$\text{\hspace{0.17em}}\left(x+1\right)\left(x-1\right)=0,$ which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.
Another example is the difference of squares formula,
$\text{\hspace{0.17em}}{a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right),$ which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.
Writing the trigonometric expression as an algebraic expression
Write the following trigonometric expression as an algebraic expression:
$\text{\hspace{0.17em}}2{\mathrm{cos}}^{2}\theta +\mathrm{cos}\text{\hspace{0.17em}}\theta -1.$
Notice that the pattern displayed has the same form as a standard quadratic expression,
$\text{\hspace{0.17em}}a{x}^{2}+bx+c.\text{\hspace{0.17em}}$ Letting
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =x,$ we can rewrite the expression as follows:
$2{x}^{2}+x-1$
This expression can be factored as
$\text{\hspace{0.17em}}\left(2x-1\right)\left(x+1\right).\text{\hspace{0.17em}}$ If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ At this point, we would replace
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and solve for
$\text{\hspace{0.17em}}\theta .$
Questions & Answers
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world