Verifying the identity using double-angle formulas and reciprocal identities
Verify the identity
$\text{\hspace{0.17em}}{\mathrm{csc}}^{2}\theta -2=\frac{\mathrm{cos}(2\theta )}{{\mathrm{sin}}^{2}\theta}.$
For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.
From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See
[link] ,
[link] , and
[link] .
We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See
[link] .
Trigonometric expressions are often simpler to evaluate using the formulas. See
[link] .
The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See
[link] and
[link] .
Section exercises
Verbal
Starting with the product to sum formula
$\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}[\mathrm{sin}(\alpha +\beta )+\mathrm{sin}(\alpha -\beta )],$ explain how to determine the formula for
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta .$
Substitute
$\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ into cosine and
$\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ into sine and evaluate.
Explain two different methods of calculating
$\text{\hspace{0.17em}}\mathrm{cos}\left(\mathrm{195\xb0}\right)\mathrm{cos}\left(\mathrm{105\xb0}\right),$ one of which uses the product to sum. Which method is easier?
Explain a situation where we would convert an equation from a sum to a product and give an example.
Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example:
$\text{\hspace{0.17em}}\frac{\mathrm{sin}(3x)+\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1.\text{\hspace{0.17em}}$ When converting the numerator to a product the equation becomes:
$\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}\mathrm{sin}(2x)\mathrm{cos}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1$
Questions & Answers
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.