Any trigonometric function whose argument is
${90}^{\circ}\pm \theta $ ,
${180}^{\circ}\pm \theta $ ,
${270}^{\circ}\pm \theta $ and
${360}^{\circ}\pm \theta $ (hence
$-\theta $ ) can be written simply in terms of
$\theta $ . For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of
${90}^{\circ}$ . From this we may expect that
$sin({90}^{\circ}+\theta )=cos\theta $ .
Function values of
${180}^{\circ}\pm \theta $
Investigation : reduction formulae for function values of
${180}^{\circ}\pm \theta $
Function Values of
$({180}^{\circ}-\theta )$
In the figure P and P' lie on the circle with radius 2. OP makes an angle
$\theta ={30}^{\circ}$ with the
$x$ -axis. P thus has coordinates
$(\sqrt{3};1)$ . If P' is the reflection of P about the
$y$ -axis (or the line
$x=0$ ), use symmetry to write down the coordinates of P'.
Write down values for
$sin\theta $ ,
$cos\theta $ and
$tan\theta $ .
Using the coordinates for P' determine
$sin({180}^{\circ}-\theta )$ ,
$cos({180}^{\circ}-\theta )$ and
$tan({180}^{\circ}-\theta )$ .
From your results try and determine a relationship between the function values of
$({180}^{\circ}-\theta )$ and
$\theta $ .
Function values of
$({180}^{\circ}+\theta )$
In the figure P and P' lie on the circle with radius 2. OP makes an angle
$\theta ={30}^{\circ}$ with the
$x$ -axis. P thus has coordinates
$(\sqrt{3};1)$ . P' is the inversion of P through the origin (reflection about both the
$x$ - and
$y$ -axes) and lies at an angle of
${180}^{\circ}+\theta $ with the
$x$ -axis. Write down the coordinates of P'.
Using the coordinates for P' determine
$sin({180}^{\circ}+\theta )$ ,
$cos({180}^{\circ}+\theta )$ and
$tan({180}^{\circ}+\theta )$ .
From your results try and determine a relationship between the function values of
$({180}^{\circ}+\theta )$ and
$\theta $ .
Investigation : reduction formulae for function values of
${360}^{\circ}\pm \theta $
Function values of
$({360}^{\circ}-\theta )$
In the figure P and P' lie on the circle with radius 2. OP makes an angle
$\theta ={30}^{\circ}$ with the
$x$ -axis. P thus has coordinates
$(\sqrt{3};1)$ . P' is the reflection of P about the
$x$ -axis or the line
$y=0$ . Using symmetry, write down the coordinates of P'.
Using the coordinates for P' determine
$sin({360}^{\circ}-\theta )$ ,
$cos({360}^{\circ}-\theta )$ and
$tan({360}^{\circ}-\theta )$ .
From your results try and determine a relationship between the function values of
$({360}^{\circ}-\theta )$ and
$\theta $ .
It is possible to have an angle which is larger than
${360}^{\circ}$ . The angle completes one revolution to give
${360}^{\circ}$ and then continues to give the required angle. We get the following results:
Investigation : reduction formulae for function values of
${90}^{\circ}\pm \theta $
Function values of
$({90}^{\circ}-\theta )$
In the figure P and P' lie on the circle with radius 2. OP makes an angle
$\theta ={30}^{\circ}$ with the
$x$ -axis. P thus has coordinates
$(\sqrt{3};1)$ . P' is the reflection of P about the line
$y=x$ . Using symmetry, write down the coordinates of P'.
Using the coordinates for P' determine
$sin({90}^{\circ}-\theta )$ ,
$cos({90}^{\circ}-\theta )$ and
$tan({90}^{\circ}-\theta )$ .
From your results try and determine a relationship between the function values of
$({90}^{\circ}-\theta )$ and
$\theta $ .
Function values of
$({90}^{\circ}+\theta )$
In the figure P and P' lie on the circle with radius 2. OP makes an angle
$\theta ={30}^{\circ}$ with the
$x$ -axis. P thus has coordinates
$(\sqrt{3};1)$ . P' is the rotation of P through
${90}^{\circ}$ . Using symmetry, write down the coordinates of P'. (Hint: consider P' as the reflection of P about the line
$y=x$ followed by a reflection about the
$y$ -axis)
Using the coordinates for P' determine
$sin({90}^{\circ}+\theta )$ ,
$cos({90}^{\circ}+\theta )$ and
$tan({90}^{\circ}+\theta )$ .
From your results try and determine a relationship between the function values of
$({90}^{\circ}+\theta )$ and
$\theta $ .
Complementary angles are positive acute angles that add up to
${90}^{\circ}$ . e.g.
${20}^{\circ}$ and
${70}^{\circ}$ are complementary angles.
Sine and cosine are known as
co-functions . Two functions are called co-functions if
$f\left(A\right)=g\left(B\right)$ whenever
$A+B={90}^{\circ}$ (i.e.
$A$ and
$B$ are complementary angles). The other trig co-functions are secant and cosecant, and tangent and cotangent.
The function value of an angle is equal to the co-function of its complement (the co-co rule).
and likewise for
$cos(\theta -{90}^{\circ})=sin\theta $
Summary
The following summary may be made
second quadrant
$({180}^{\circ}-\theta )$ or
$({90}^{\circ}+\theta )$
first quadrant
$\left(\theta \right)$ or
$({90}^{\circ}-\theta )$
$sin({180}^{\circ}-\theta )=+sin\theta $
all trig functions are positive
$cos({180}^{\circ}-\theta )=-cos\theta $
$sin({360}^{\circ}+\theta )=sin\theta $
$tan({180}^{\circ}-\theta )=-tan\theta $
$cos({360}^{\circ}+\theta )=cos\theta $
$sin({90}^{\circ}+\theta )=+cos\theta $
$tan({360}^{\circ}+\theta )=tan\theta $
$cos({90}^{\circ}+\theta )=-sin\theta $
$sin({90}^{\circ}-\theta )=sin\theta $
$cos({90}^{\circ}-\theta )=cos\theta $
third quadrant
$({180}^{\circ}+\theta )$
fourth quadrant
$({360}^{\circ}-\theta )$
$sin({180}^{\circ}+\theta )=-sin\theta $
$sin({360}^{\circ}-\theta )=-sin\theta $
$cos({180}^{\circ}+\theta )=-cos\theta $
$cos({360}^{\circ}-\theta )=+cos\theta $
$tan({180}^{\circ}+\theta )=+tan\theta $
$tan({360}^{\circ}-\theta )=-tan\theta $
These reduction formulae hold for any angle
$\theta $ . For convenience, we usually work with
$\theta $ as if it is acute, i.e.
${0}^{\circ}<\theta <{90}^{\circ}$ .
When determining function values of
${180}^{\circ}\pm \theta $ ,
${360}^{\circ}\pm \theta $ and
$-\theta $ the functions never change.
When determining function values of
${90}^{\circ}\pm \theta $ and
$\theta -{90}^{\circ}$ the functions changes to its co-function (co-co rule).
Function values of
$({270}^{\circ}\pm \theta )$
Angles in the third and fourth quadrants may be written as
${270}^{\circ}\pm \theta $ with
$\theta $ an acute angle. Similar rules to the above apply. We get
third quadrant
$({270}^{\circ}-\theta )$
fourth quadrant
$({270}^{\circ}+\theta )$
$sin({270}^{\circ}-\theta )=-cos\theta $
$sin({270}^{\circ}+\theta )=-cos\theta $
$cos({270}^{\circ}-\theta )=-sin\theta $
$cos({270}^{\circ}+\theta )=+sin\theta $
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 .
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2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
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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.