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
$\text{\hspace{0.17em}}\theta .$
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
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Mahi
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How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it