Now going for clockwise
When we put a random angle from quadrant 1, the range of -360<θ<-270, -278° in trial to verify the conjecture, the values of sin, cos and tan turn out to be positive.
When we put a random angle from quadrant 2, the range of -270<θ<-180, -204° in trial to verify the conjecture, the value of sin turn out to be positive while the values of cos and tan turn out to be negative.
When we put a random angle from quadrant 3, the range of -180<θ<-90, -164° in trial to verify the conjecture, the value of tan turn out to be positive while the values of sin and cos turn out to be negative.
When we put a random angle from quadrant 4, the range of -90<θ<0, -7° in trial to verify the conjecture, the value of cos turns out to be positive and the values of sin and tan turn out to be negative.
Sin θ= opposite/ hypotenuse= O/H= y/r
Cos θ= adjacent/ hypotenuse= A/H= x/r
Tan θ= opposite/ adjacent=O/A= y/x
Since r should always be positive as what I said in the top. Also by this equation, this is also the another step to find out if it is positive or negative.
Quadrant 1—( +,+) sin θ=y/r =+
cos θ=x/r=+
tan θ=y/x= +
Quadrant I
The value of y equals a positive number in the quadrant 1 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of y is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of x equals a positive number in the quadrant 1 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of x is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of y and the value of x equal to positive numbers respectively in quadrant 1. When the value of y is divided by the value of x, a positive number is divided by a positive number resulting to a positive number.
Quadrant 2—(-,+) sin θ=y/r =-
cos θ=x/r=-1
tan θ=y/x= -
Quadrant II
The value of y equals a positive number in the quadrant 2 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of y is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of x equals a negative number in the quadrant 2 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of x is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of y equals a positive number and the value of x equals to a negative number in quadrant 2. When the value of y is divided by the value of x, a positive number is divided by a negative number resulting to a negative number.
Quadrant 3—(-,-) sin θ=y/r =-
cos θ=x/r=-
Quadrant III
The value of y equals a negative number in the quadrant 3 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of y is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of x equals a negative number in the quadrant 3 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of x is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of y and the value of x equal to a negative number respectively in quadrant 3. When the value of y is divided by the value of x, a negative number is divided by a negative number resulting to a positive number.
Quadrant IV
The value of y equals a negative number in the quadrant 4 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of y is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of x equals a positive number in the quadrant 4 and the value of r equals a positive number as mentioned beforehand. Likewise, when the value of x is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of y equals a negative number and the value of x equals a positive number in quadrant 4. When the value of y is divided by the value of x, a negative number is divided by a positive number resulting to a negative number.
Axis: The signs of the coordinates on the positive x axis are (positive,0); the value of x equals a positive number and the value of y equals 0.
Positive x axis--- (+, 0) sin θ=y/r =+
cos θ=x/r=0
tan θ=y/x= 0
Negative x axis---(-,0) sin θ=y/r =-
cos θ=x/r=0
tan θ=y/x= 0
Positive y axis---(0,+) sin θ=y/r =+
cos θ=x/r=0
tan θ=y/x= undefined
Negative y axis---(0,-) sin θ=y/r =-
cos θ=x/r=0
tan θ=y/x= undefined
The value of x equals a positive number on the positive x axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of x is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of y equals 0 on the positive x axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of y is divided by the valued of r, 0 is divided by a positive number resulting to 0.
The value of y equals 0 and the value of x equals a positive number on the positive x axis. Therefore, when the value of y is divided by the value of x, 0 is divided by a positive number, resulting to 0.
The signs of the coordinates on the negative x axis are (negative,0); the value of x equals a negative number and the value of y equals 0.
The value of x equals a negative number on the negative x axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of x is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of y equals 0 on the negative x axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of y is divided by the value of r, 0 is divided by a positive number resulting to 0.
The value of y equals 0 and the value of x equals a positive number on the negative x axis. Therefore, when the value of y is divided by the value of x, 0 is divided by a positive number, resulting to 0.
The signs of the coordinates on the positive y axis are (0,positive); the value of x equals 0 and the value of y equals a positive number.
The value of y equals a positive number on the positive y axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of y is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.
The value of x equals 0 on the positive y axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of x is divided by the value of r, 0 is divided by a positive number resulting to 0.
The value of y equals a positive number and the value of x equals 0 on the positive y axis. Therefore, when the value of y is divided by the value of x, a positive number is divided by 0, resulting to undefined value.
The signs of the coordinates on the negative y axis are (0, negative); the value of x equals 0 and the value of y equals a negative number.
The value of y equals a negative number on the negative y axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of y is divided by the value of r, a negative number is divided by a positive number resulting to a negative number.
The value of x equals 0 on the negative y axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of x is divided by the value of r, 0 is divided by a positive number resulting to 0.
The value of y equals a negative number and the value of x equals 0 on the negative y axis. Therefore, when the value of y is divided by the value of x, a positive number is divided by 0, resulting to undefined value.
The conjecture informally by considering further examples of sin θ, cos θ, and tan θ that are not in the table of values, in quadrant 1 that have not been mentioned in the table of values are positive, all in Quadrant 2 only have the sine value as positive, all in Quadrant 3 have only the tangent value as positive, and in Quadrant 4 only the cosine value is positive.
Sin θ= opposite/ hypotenuse= y/r
Cos θ= adjacent/ hypotenuse= x/r
Tan θ= opposite/ adjacent= y/x
Since r should always be positive as what I said in the top.
Getting the sine value for theta, the length of the side opposite to theta is divided by the length of the hypotenuse or opposite side of theta divided by hypotenuse which is r. Thus, in this investigation, . Similarly, getting the cosine of any value of θ, wherein the length of the adjacent side to θ is divided by the length of the hypotenuse, would be. Attaining the tangent, in which the length of the opposite side is divided by the length of the side adjacent to θ, would have the equation .
Now by step by step, the conjecture will be described in each step.
First, in first quadrant where (00 to 900, as well as -2700 to -3600) all values of y are positive, radius which is always positive or in other words real number, dividing the y value or opposite of that will always give positive quotient.
Since cosine is also x/r or adjacent divided by hypotenuse R. It will become positive again which the same situation of Sin θ.
In first Quadrant Tangent opposite/hypotenuse will also be positive in quadrant 1 ().
In Quadrant 2, the y value is negative or meaning that sin θ is negative, but radius and x remaining positive. Influencing the sine and tangent value of θ. However by the y value being negative it will not only affect sin θ but also tan θ= y/x, as a negative number divided by a positive results in a negative answer.
As it can be seen in the diagram, all x and y values in Quadrant 3 are negative, though the r value remains positive as it is a radius of a circle, which cannot be negative. As all three formulas for calculating the sine, cosine and tangent values involve x and y, all are affected. It is given that and . Because the negative values of x and y are being divided by the positive value of r, the result will always be negative.
However, in the tangent formula, wherein, a negative value is being divided by another negative value, which will in turn give us a positive quotient.
In Quadrant 4, as can be referred to in the diagram on the right, only the x value is negative, whilst the values for y and r are positive. This in turn affects the formulas for attaining the cosine and tangent of θ, as they are the formulas which involve the x value. In and , the negative x value is either being divided by or is dividing a positive value, which would result in a negative quotient. However, because and both y and r values are in this case positive, the sine of θ in Quadrant 4 will always be positive.
Part B: Trigonometric Identities
Upon the analysis of the table of values from Part A, we can conjecture the relationship among sinθ, cosθ, and tanθ for any angle θ.
As can be clearly seen in the table above, the values for tanθ and are identical.
Further to prove this, I used TI 83 calculator for better explanation such as random numbers.
First for quadrant 1, I will choose a value or a degree for tangent theta between 0-90 degrees and -360 to -270. Which are 76 and -298 degrees.
For other revolutions
For quadrant two the range between 90 to 180 and -270 to -180
With the example of 127 and -222
For quadrant 3, the range of tan theta will be 180-270 and -180 to -90.
With the example of 199 and -101
Finall for quadrant four with the range of 270 to 360 and -90 to 0
With the example of 354 and -4 degrees.
Therefore, through this further examples we again can see that the conjecture between tanθ and are identical.
a) Cos θ= x/r
sin θ= y/r
tan θ= y/x
sin θ/ Cos θ=(y/r)/(x/r) so y/r divided by x/r equals y/r times r/x. Which r gets simplified equaling y/x. So if we see tan it will become y/x which means sin θ/ Cos θ equals to tan θ.
Equation:
b) Expressing Cos θ,tan θ and tan θ in terms of x,y and r
then for x value it will be x= Cos θ( r ), y value y= sin θ( r ) and tan equaling Tan θ= y/x equaling sin θ( r ) divided by Cos θ( r ) then as what I said in the above, the r gets simplified just leaving.
y= sin θ( r )
Tan θ= y/x equaling sin θ( r ) divided by Cos θ( r ) then as what I said in the above, the r gets simplified just leaving
sin²θ+cos²θ, the relationship between sin²θ and cos²θ is whenever I add them sin²θ+cos²θ this becomes one and even though whatever angle we put for θ the sum sin²θ+cos²θ will always be 1.
In Part A when expressing sin θ, cos θ and tan θ in terms of x, y and r. Which sin θ=y/r, cos θ= x/r and cos² θ= x²/r² so when squaring cosine theta and squaring sine theta sin² θ will equal= y²/r² after this when adding sin² θ + cos² θ this will equal x²/r² + y²/r² which the sum is x²+ y²/r² then since x ² + y² is r² we will substitute r² in the equation of x²+ y²/r² giving r²/ r² simplifying to 1. As a result x²+ y²=r² so in number 1 Part A
sin² θ + cos² θ= 1.
The value of Sin θ and Cos θ in the first quadrant 0<= θ <= 90. The value of sine theta equals the value of y divided by the value of r and the value of cosine theta equals the value of x divided by the value of r. Therefore, the square of the value of sine theta equals the square of the value of y divided by the square of the value of r and the square of the value of cosine theta equals the square of the value of x divided by the square of the value of r. When the square of the value of cosine theta is added to the square of the value of sine theta, the result of the square of the value of x divided by the square of the value of r is added to the result of the square of the value of y divided by the square of the value of r.
=() + ()
=
=
=1
The values of sinθ and cosθ within the range of 0°≤θ≤90° are analyzed in order to conjecture another relationship between sinθ and cosθ for any angle θ.
A portion of Table 1 within the range of 0°≤θ≤90° showing the relationship between sinθ and cosθ for any angle θ
The colors matching, by twos are meaning that they share the same values. Based on this table we can see that there is relation between sinθ and cosθ from 0 to 90 degree. The example of sin30 equals to cos60 which the relation is that the theta is same. So the value of sinθ is equal to cos (90-θ) then next step if we determine sinx= cosy. Then x+y will result 90 degree this tells us that it leads to complementary angle. As shown in the colored portions of the table above, sin(0) equals cos(90), sin(10) equals cos(80), sin(20) equals cos(70), sin(30) equals cos(60), sin(40) equals cos(50), sin(50) equals cos(40), sin(60) equals cos(30), sin(70) equals cos(20), sin(80) equals cos(10), and sin(90) equals cos(0).
Therefore, when the angle of sine and the angle of cosine are summed up, it is equal to 90; complementary.
sinx=cosy
x+y=90
Thus, the value of sine theta is equal to the value of cosine 90 minus theta.
sin θ=cos(90- θ)
Again to verify the conjecture, two random angles that are not already in the table of balues are tested from the first quadrant in the range of 0°≤θ≤90°.
When 36 is to represent the value of x and 54 is to represent the value of y in the conjecture sinx=cosy, sin(36) would equal to sin(54) Again this should add up to 90 degrees.
sinx=cosy
sin(36)=cos(54)
sin θ=cos(90- θ)
sin(36)=cos(90-36)
0.5878=0.5878
When 13 is to represent the value of x and 77 is to represent the value of y in the conjecture sinx=cosy, sin(13) would equal to cos(77). Again this should add up to 90 degrees.
sinx=cosy
sin(13)=cos(77)
sin θ=cos(90- θ)
sin(13)=cos(90-13)
0.2250=0.2250
When 54 is to represent the value of x and 36 is to represent the value of y in the conjecture sinx=cosy, sin(54) would equal to sin(36) Again this should add up to 90 degrees.
sinx=cosy
sin(54)=cos(36)
sin θ=cos(90- θ)
sin(54)=cos(90-54)
0.8090=0.8090
When 77 is to represent the value of x and 13 is to represent the value of y in the conjecture sinx=cosy, sin(77) would equal to sin(13). Again this should add up to 90 degrees.
sinx=cosy
sin(77)=cos(13)
sin θ=cos(90- θ)
sin(77)=cos(90-77)
0.9744=0.9744
Considering a triangle in the first quadrant with angle θ, the measure of the other acute angle in the triangle in terms of θ is (90-θ). The conjecture sin θ=cos(90- θ) is to be proved by using the values of sine and cosine in terms of x,y,r of the angle (90-θ).
The origin and a radius of r units drawn to some point in the four quadrant of the circle forming a right triangle with its sides x,y, and r and its acute angles θ and (90- θ)
With a triangle in the graph above labeled with x,y and r as its sides, the value of cosθ equals to the adjacent side over the radius which means x/r. The value of cosθ is also proved to equal the value of sin(90-θ) which equals to the opposite side over the radius meaning x over r also. As cosθ= and sin(90-θ)= , therefore, cosθ= sin(90-θ).
So, the value of sinθ equals to the opposite side over the radius meaning y over r. This equals to the value of cos(90-θ) which means the adjacent side over the radius resulting y over r also. As sinθ= and cos(90-θ)= , therefore, cos(90-θ)= sinθ.
Part C: Sine and Cosine Graphs
The domains -≤θ≤ or -6.28≤ θ≤6.28 is used in the domains of360°≤θ≤360° is expressed in radians as≈6.28.
The terms maxima, minima, amplitude, period and frequency are described in the graphs of y=sinθ and y=cosθ for the domains -≤θ≤.
The sinθ graph passes through the points (-,0), (-,1), (-,0), (-,-1), (0,0), (,1), (,0), (,-1), and (,0) with 9 coordinates as seen in the graph above.
Upon the analysis of the pattern of the graph,
As the value of θ increases from -to -, sinθ goes from 0 to 1.
As the value of θ increases from - to -, sinθ goes from 1 to 0.
The value of θ increases from - to -, sinθ goes from -1 to 0.
The value of θ increases from 0 to , sin θ goes from 0 to 1.
The value of θ increases from to , sinθ goes from 1 to 0.
The value of θ increases from to , sinθ goes from 0 to -1.
The value of θ increases from to , sinθ goes from -1 to 0.
The maxima, the highest point in the y value, of the graph is 1 and the minima, the lowest point in the y value, of the graph is -1. The amplitude is the distance of the maximum y value to the middle y value, which in this graph shows, resulting to 1. The period refers to x values, how long it takes for the pattern to begin again. The period in the graph is which approximately 6.28 when expressed to radian are. The frequency is how many cycles or pattern the graph have gone through. When expressed to radian the frequency is the reciprocal of the period meaning which is approximately 0.1592.
In this cosθ graph the line passes through the points (-,1), (-,0), (-,-1), (-,0), (0,1), (,0), (,-1), (,0), and (,1) again with 9 coordinates.
Upon the analysis of the pattern of the graph,
As the value of θ increases from - to -, cosθ goes from 1 to 0.
As the value of θ increases from - to -, cosθ goes from 0 to -1.
As the value of θ increases from - to -, cosθ goes from -1 to 0.
As the value of θ increases from 0 to, cosθ goes from 1 to.
As the value of θ increases from to, cosθ goes from 0 to -1.
As the value of θ increases from to, cosθ goes from -1 to 0.
As the value of θ increases from to, cosθ goes from 0 to 1.
The maxima which are the maximum value for y or the highest point in y value, of this the maxima, the highest point in the y value graph is 1 and the minima, the lowest point in the y value, of the graph is -1.
For the graph is 1 and the minima, the lowest point in the y value, of the graph is -1.
The amplitude, the distance of the maximum y value to the middle y value, in the is, resulting to 1.
The relationship between the uses of x is that to see the pattern repeating.
The period in the graph is which approximately 6.28 when expressed to radian are. The frequency refers to how many cycles a graph have gone through. When expressed to radian the frequency is the reciprocal of the period meaning which is approximately 0.1592.
When referring to maxima, minima, period, amplitude and frequency, the graph and the graph are exactly the same.
The comparison of the points the graphs and pass through the 9 coordinates.
y=3cos (θ)
y= (-3)cosθ
y=3cos (-θ)
The maxima of this graph y=3cosθ is 3 and the miima is -3.
The amplitude in the y=3cosθ is resulting to 3. The period in the y=3cosθ is which is approximately 6.28 when expressed to radian. Expressing it to radian the frequency is approximately 0.1592.
y=3cos (-θ)
Reflected through the y-axis from the y=3cosθ graph so it is coincident with the y=3cosθ graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the y=3cosθ graph.
y= (-3) cosθ
Reflected through the x axis from the y=3cosθ graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the y=3cosθ graph.
y=2sinθ
y=-2sinθ
y=2sin(-θ)
Period:
The period in the y=sinθ is which is approximately 6.28 when expressed to radian. Then the frequency will be 0.1592 in terms of radian.
y=2sin(-θ)
Reflected through the y axis from the y=2sinθ graph so it is coincident with the y=2sinθ graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the y=2sinθ graph.
y=-2sinθ
Reflected through the x axis from the y=2sinθ graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the y=2sinθ graph.
y=2sinθ
The maxima of the y=2sinθ graph is 2 and the minima of the y=2sinθ graph is -2. The amplitude in the y=2sinθ isresulting to 2.
The 3 graphs above, After analyzing the 3 αsinθ, we can see that the values of a in y=αsinθ becomes the ranges the graph to be -α≤sin θ≤a instead of -1≤sinθ≤1.
Also, the amplitude of y=we can understand that the values of α in y=αsinθ sets the range of the graph to be -α≤sin θ≤a instead of -1≤sinθ≤1. Also, the amplitude of y=αsinx is the largest value of y and which given by.
Therefore, the amplitude of y=αsinx and y=αcosx will be the largest value of y and will be given by amplitude= and sets the range of the graph by giving the values of both maxima and the minima. (The curvy scribble looking line.)
The graph is reflected across the y axis from the graph so it is coincident with the graph;
Therefore, the values of maxima, minima, amplitude, period and frequency are the same as the graph.
The frequency is approximately 0.1592 when expressed to radian.
The graph is reflected across the x axis from the graph; therefore, the values of maxima, minima, amplitude,
Period and Frequency are the same as in the graph which is 0.1592.
The conjecture is verified as the value of α, giving the amplitude, maxima and minima in the graph,
thus setting the range of -≤≤.
The maxima of the graph is and the minima of the graph is -.
The amplitude in the graph is calculated by resulting to. The period in the graph is which approximately 6.28 when expressed to radian are.
Thegraph is reflected across the x axis from the graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the graph.
The graph is reflected across the y axis from the graph so it is coincident with the graph; therefore, the values of maxima, minima, amplitude, period and frequency are the same as the graph.
The maxima of the graph is and the minima of the graph is -.
The amplitude in the graph is calculated by resulting to. The period in the graph is which approximately 6.28 when expressed to radian are.
The frequency is approximately 0.1592 when expressed to radian.
Therefore, the conjecture will be the value of α, gives the amplitude, maxima and minima in the graph, thus setting the range of - ≤≤ .
y=sinbθ and y=cosbθ graphs for different values of b using the domains of -≤θ≤ are formed to see any constraints on the values of b.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is calculated by, resulting to 1. The period in the graph is which approximately 18.8496 when expressed to radian are. The frequency is approximately 0.5236 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1.
The amplitude in the graph is resulting to 1.
The period in thegraph is which is approximately 2.0944 when expressed to radian. The frequency is approximately 0.4775 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is calculated by, resulting to 1. The period in thegraph is which approximately 12.5664 when expressed to radian are. The frequency is approximately 0.7854 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is, resulting to 1.
The period in thegraph is which is approximately 3.1416 when expressed to radian.
The frequency is approximately 0.3183 when expressed to radian.
When we observe some specific features of the above graphs, we can see how the value of b affects graphs of y=sinbx and y=cosbx. The period of both y=sinbx and y=cosbx graphs is. The value of b inis 3 so it is, which is how long the cycle takes to repeat while the value of b in is so it is and when calculated, which is how long the cycle takes to repeat.
-≤bθ≤
If b is positive, the above inequality is 0≤θ≤. When 0<b<1, the period of y=sinbx is greater than and represents a horizontal stretching of the graph of y=asinx. Similarly, if b>1, the period of y=sinbx is less than and represents a horizontal compression of the graph of y=asinx. When b is negative, the above inequality becomes ≤θ≤0.
For either a positive or negative value of b, one cycle (period) of the graph of y=sinbθ and y=cosbθ are obtained respectively on an interval of. Then, the frequency, the reciprocal of the period, is.
The conjecture is now verified by considering further examples of b.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is calculated by resulting to 1. The period in thegraph is which approximately 3.6276 when expressed to radian are. The frequency is which is approximately 0.2757 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is calculated by resulting to 1. The period in thegraph is which approximately 10.8828 when expressed to radian are. The frequency is which is approximately 0.9069 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1. The amplitude in the graph is calculated by resulting to 1. The period in thegraph is which approximately 4.4429 when expressed to radian are.
The frequency is which is approximately 0.5642 when expressed to radian.
The maxima of the graph is 1 and the minima of the graph is -1.
The amplitude in the graph is calculated by resulting to 1.
The period in thegraph is which is approximately when expressed to radian. The frequency is which is when expressed to radian.
Conclusion:
