The Sine Function
Pictured above is the sine function graphed as a trigonometric function. This is obtained from the sine values versus its angles. However, unlike the unit circle, the sine angles are not measred in radians (in units of pi), but in degrees. Just to remind you, radians are converted to degrees by multiplying the current radian by 180/pi. To convert degrees to radians, multiply the current degree by pi/180.
The period, or the distance of one complete cycle, of the sine graph is 2pi. The amplitude of the parent sine function is 1, as shown. The amplitude is the distance between the center line to the maximum point on the y-axis(or the center line to the minimum point--they should be the same).
Just to give you more of an idea of what the sine function looks like (more than the limited graph shown here), below is a table of the sin values versus its angles:
| Theta (the sine angle) | Sine theta degrees |
| 0 | 0.0 |
| 30 | 0.5 |
| 45 | 0.71 |
| 60 | 0.87 |
| 90 | 1.0 |
The Cosine Function
Shown above is the cosine curve graphed as a trigonometric function. Like the sine curve, it has a period of 2pi, and its parent function has an amplitude of 1. However, unlike the sine curve, it begins its cycle at the point (0, 1), whereas the sine curve begins its cycle at the point (0,0).
Just to give you more of an idea of what the sine function looks like (more than the limited graph shown here), below is a table of the sin values versus its angles:
| Theta (the cosine angle) | Cosine theta degrees |
| 0 | 1.0 |
| 30 | 0.87 |
| 45 | 0.71 |
| 60 | 0.5 |
| 90 | 0.0 |
Translations
The sine and cosine graphs viewed above are the parent functions. However, these graphs can be altered by changing their amplitudes, periods, and shifting them upwards or downwards. The parent functions of the sine and cosine curves can otherwise be written as the equations of y = a sin(x) + c, and y = a cos(x)+ c, where the variables 'a', 'b', and 'c' are all '1'. The 'a' variable represents the amplitudes of the graphs, 'b' which is the coefficient of 'x' represents the periods of the graphs, and 'c' represents the translation of the entire graph in regards to the y-axis, which, in the parent functions, is zero.
Some problems to help you better understand...
'a' is '2', 'b' is '2pi/2' or 'pi' and 'c' is '1 unit' which will shift the entire graph upwards. (You may have noticed that 'b' was calculated differently from 'a' and 'c'. Unlike 'a' and 'c', the period is not simply a number represented by a variable. 'b' actually represents 360degrees/'b', or 2pi/'b'.)
The parent cosine graph is reflected across the x-axis, due to the negative sign in the new equation; the period is now 4pi [remember...2pi/b, or in this case, (2pi/(1/2)), thus the period of 4pi]; and translating the parent function upwards by 2 units.
People that go in or out of harbor need to know the behaviors of the ocean's tide in order to determine whether it is safe to leave or come in to harbor. The tide is caused by gravitational pulls from the sun and moon and earth's rotation. To predict the behavior of tides, the behaviors must first be recorded, thus allowing people to guess what the tide's next behavior will be. What is the connection? Patterns of tide behaviors is just like sine and cosine graphs, as shown below of tide behaviors in Bombay over a period of 14 days in January.
Lesson 3.4
Geometry of Three Dimensional Shapes
The distance between any two points (x,y,z) and (x',y',z') is given by the distance formula of d(P,P')=((x'-x)^2+(y'-y)^2+(z'-z)^2).
For example: Find the distance between the point (-1,-3,1) and (3,4,-2).
d(P,P')=((x'-x)^2+(y'-y)^2+(z'-z)^2
d(P,P')=((3+1)^2+(4+3)^2+(-2-1)^2=>74
To find the length of a line joining a midpoint with a midpoint,use the formula ((x+x')/2, (y+y')/2, (z+z')/2).
For example: Find the midpoint of the line segment from (2,-3,6) to (3,4,-2).
Using the formula, the answer will come out to be...((2+3)/2,(-3+4)/2,(6+(-2))/2) which will equal (5/2,1/2,2).
Lesson 3.5
Vectors
Vectors as Displacements in the Plane
When referring to vectors on a plane, they are expressed in terms of what is known as a basis. The most commonly used basis is vectors 'i' and 'j'. These two vectors can be used to define any vector lying in the plane of the page. Therefore, if vector 'a' is 3i+2j, it implies that from the original point, the vector will travel over 3 units with regards to the x-axis, and up 2 units with regards to the y-axis. Another example, vector 'b' being -i-5j, would imply from the original point, traveling to the left 1 unit, down 5 units. However, vectors may be written as a column vector. For example, vector 'c'=2i+3j may be written as:
(2
3)
Components of a Vector
The components of a vector are its 'i' and 'j' vectors. For example, if vector'd' = -i+3j, then -i and 3j would be its components.
Vector Arithmetic
Vectors are added in the same manner as algebraic terms, where only like terms can be added or subtracted. For example: if vector'a'=2i-j and vector'b'=-i+3j, then 'a'+'b'=2i+j+(-i+3j)=>i+2j. To find the difference between vectors 'a' and 'b', 'a'-'c'=(2i+j)-(-i+3j)=>3i-2j. (However, there is a difference between finding the differences between two vectors, and adding a negative vector. Finding the difference between two vectors was just shown, and is simply subtracting one vector from the other. However, adding a negative vector is an entirely different story. So be careful to differentiate between the two.)
The zero vector is expressed by 0=(0,0)
The negative vector is expressed by -'a'=-(a,a')=(-a,-a')
The vector multiplication theorem implies that the length of a directed line segment that represents two vectors, e.g. 'a' and 'b', that 'a' will be 'b' times the lenght of the directed line segment 'a'. Some properties of scalar multiples of vectors:
c(a+b)=ca+cb
(b+c)a=ba+ca
(bc)a=b(ca)=c(ba)
1a=a
0a=0
The length of a vector is sometimes known as its magnitude (or absolute value). To determine a vector's magnitude, the vector's components must be squared, added, and then the square root of the sum of squared components is the magnitude of the vector. It is the extended version of the Pythagorean theorem. For example, a vector of -i+2j-5k,will be ((-1)^2+2^2+(-5^2)), which equals 30.
LINKS
Hope you learned valuable trig facts!!
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