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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. |
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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.
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.
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