# 7.2 The trig functions for any angle and applications

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## The trig functions for any angle

So far we have defined the trig functions using right angled triangles. We can now extend these definitions to any angle. We do this by noting that the definitions do not rely on the lengths of the sides of the triangle, but only on the angle. So if we plot any point on the Cartesian plane and then draw a line from the origin to that point, we can work out the angle of that line. In [link] points P and Q have been plotted. A line from the origin to each point is drawn. The dotted lines show how we can construct right angle triangles for each point. Now we can find the angles A and B.

You should find the angle A is $63,{43}^{\circ }$ . For angle B, you first work out x ( $33,{69}^{\circ }$ ) and then B is ${180}^{\circ }-33,{69}^{\circ }=146,{31}^{\circ }$ . But what if we wanted to do this without working out these angles and figuring out whether to add or subtract 180 or 90? Can we use the trig functions to do this? Consider point P in [link] . To find the angle you would have used one of the trig functions, e.g. $\mathrm{tan}\phantom{\rule{1pt}{0ex}}\theta$ . You should also have noted that the side adjacent to the angle was just the x-co-ordinate and that the side opposite the angle was just the y-co-ordinate. But what about the hypotenuse? Well, you can find that using Pythagoras since you have two sides of a right angled triangle. If we were to draw a circle centered on the origin, then the length from the origin to point P is the radius of the circle, which we denote r. Now we can rewrite all our trig functions in terms of x, y and r. But how does this help us to find angle B? Well, we know that from point Q to the origin is r, and we have the co-ordinates of Q. So we simply use the newly defined trig functions to find angle B! (Try it for yourself and confirm that you get the same answer as before.) One final point to note is that when we go anti-clockwise around the Cartesian plane the angles are positive and when we go clockwise around the Cartesian plane, the angles are negative.

So we get the following definitions for the trig functions:

$\begin{array}{ccc}\hfill sin\theta & =& \frac{x}{r}\hfill \\ \hfill cos\theta & =& \frac{y}{r}\hfill \\ \hfill tan\theta & =& \frac{y}{x}\hfill \end{array}$

But what if the x- or y-co-ordinate is negative? Do we ignore that, or is there some way to take that into account? The answer is that we do not ignore it. The sign in front of the x- or y-co-ordinate tells us whether or not sin, cos and tan are positive or negative. We divide the Cartesian plane into quadrants and then we can use [link] to tell us whether the trig function is positive or negative. This diagram is known as the CAST diagram.

We can also extend the definitions of the reciprocals in the same way:

$\begin{array}{ccc}\hfill cosec\theta & =& \frac{r}{x}\hfill \\ \hfill sec\theta & =& \frac{r}{y}\hfill \\ \hfill cot\theta & =& \frac{x}{y}\hfill \end{array}$

Points R(-1;-3) and point S(3;-3) are plotted in the diagram below. Find the angles $\alpha$ and $\beta$ .

1. We have the co-ordinates of the points R and S. We are required to find two angles. Angle $\beta$ is positive and angle $\alpha$ is negative.

2. We use tan to find $\beta$ , since we are only given x and y. We note that we are in the third quadrant, where tan is positive.

$\begin{array}{ccc}\hfill tan\left(\beta \right)& =& \frac{y}{x}\hfill \\ \hfill tan\left(\beta \right)& =& \frac{-3}{-1}\hfill \\ \beta & =& {tan}^{-1}\left(3\right)\hfill \\ \beta & =& 71,{57}^{\circ }\hfill \end{array}$
3. We use tan to calculate $\alpha$ , since we are only given x and y. We also note that we are in the fourth quadrant, where tan is positive.

$\begin{array}{ccc}\hfill tan\left(\alpha \right)& =& \frac{y}{x}\hfill \\ \hfill tan\left(\alpha \right)& =& \frac{-3}{3}\hfill \\ \alpha & =& {tan}^{-1}\left(-1\right)\hfill \\ \alpha & =& {-45}^{\circ }\hfill \end{array}$
4. Angle $\alpha$ is ${-45}^{\circ }$ and angle $\beta$ is $71,{57}^{\circ }$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? 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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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