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Key-value pairs

Figure 4 contains the text values associated with each of the Braille keys.

Figure 4 . Text values for Braille keys in file Phy1020b2svg.
m: A 3-4-5 Triangle n: 4o: Vertical axis p: 0q: 0 r: Adjacent sides: 53.13 Degrees t: adju: 3 v: oppw: Opposite side x: hypy: Hypotenuse z: Horizontal axisA: Not drawn to scale

The length of the hypotenuse

Now that you have your right triangle on the graph board, or you have access to tactile graphics created from the svg file, and you know thelengths of the adjacent and opposite sides, do you remember how to calculate the length of the hypotenuse?

The Pythagorean theorem

Hopefully you know that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two othersides. Thus, the length of the hypotenuse is equal to the square root of the sum of the squares of the other two sides.

In this case we can do the arithmetic in our heads to compute the length of the hypotenuse. (I planned it that way.)

The square of the adjacent side is 9. The square of the opposite side is 16. The sum of the squares is 25, and the square root of 25 is5. Thus, the length of the hypotenuse is 5.

A 3-4-5 triangle

You have created a rather unique triangle. You have created a right triangle in which the sides are either equal to, or proportional to the integervalues 3, 4, and 5.

I chose this triangle on purpose for its simplicity. We will use it to investigate some aspects of trigonometry.

The sine and arcsine of an angle

You will often hear people talk about the sine of an angle or the cosine of an angle. Just what is the sine of an angle anyway?

Although the sine of an angle is based on very specific geometric considerations involving circles (see (External Link) ), for our purposes, the sine of an angle is simply a ratio between the lengths of two different sides of a righttriangle.

A ratio of two sides

For our purposes, we will say that the sine of an angle is equal to the ratio of the opposite side and the hypotenuse. Therefore, in the case of the 3-4-5 triangle that youhave on your graph board, the sine of the angle at the origin is equal to 4/5 or 0.8.

If we know the lengths of the hypotenuse and the opposite side, we can compute the sine and use it to determine the valueof the angle. (We will do this later using the arcsine.)

Conversely, if we know the value of the angle but don't know the lengths of the hypotenuse and/or the opposite side, we can obtain the value of the sine of theangle using a scientific calculator (such as the Google calculator) or lookup table.

The sine of an angle -- sample computation

Enter the following into the Google search box:

sin(53.13010235415598 degrees)

The following will appear immediately below the search box:

sin(53.13010235415598 degrees) = 0.8

This matches the value that we computed above as the ratio of the opposite side and the hypotenuse.

The arcsine (inverse sine) of an angle

The arcsine of an angle is the value of the angle having a given sine value. In other words, if you know the value of the sine of an unknown angle, you canuse a scientific calculator or lookup table to find the value of the angle.

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
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Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Ramkumar Reply
what is nano technology
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what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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