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
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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?