Math Honors 1: Trigonometry Investigation Task

This investigation will present an analysis on initial problem by setting patterns and establishing mathematical relationships between the parameters in the problem. In this specific investigation, I will find to see the relationship between radius R and point X and Y in a coordinate plane. The center of the circle will be (0, 0) or the origin and the radius R will be unknown. Point P with the coordinate (x,y) will always be on the circumference of the circle, and will always be perpendicular from the X axis to the point.

Part A: Circle Trigonometry

The diagram above, the radius r and the point (x, y) will form a right triangle. Therefore we can state that the equation to find the relationship can be described as Pythagorean Theorem. Wherever the coordinates (x, y) lies on the circle it will always form right triangle.

Therefore we can use the equation distance between the two points.

After we can state that the endpoints of the radius are set in the origin meaning that one end point will always set in the origin (0, 0). So we can simplify our equation further more.

Then after squaring this will result in Pythagorean Theorem

The diagram above with point R and Q is just to back up or example the idea of relationship use of right triangle to use the equation of Pythagorean Theorem.

As a result, due to Pythagorean Theorem, r squared will always be positive. And by using the relation equation the x and y value can be positive and negative thus 0. Negative and positive values meaning that they can be in any exact quadrants but one endpoint being at origin creating a right triangle. So we can see that the radius can’t ever be zero or negative it should always be positive or in other words real number. By using theta, if it goes counter clock wise it will become positive while going in the direction of clockwise it becomes negative. Further more, based on our knowledge on Math Honor 1, we always know that hypotenuse in right triangles are the longest of among three sides. And thus since R is the hypotenuse, as explained before x and y can be any number as long as it is less than the radius. Therefore leading radius R always positive and real number. Thus as logical expression the circle’s radius can’t be represented as a negative value. Plus Radius is fixed value.

Since we are working on counter two revolutions- counter clockwise, clock wise on the circle, positive angles are measured in a counter clock wise direction and negative angles are measure in clockwise direction. First, for counter clock wise direction where the theta is positive the 0º is on the positive x axis. Then theta will be the only constraint which is 0º ≤ θ≤ 360º.

Used by Excel 2007, the table of values of theta in comparing in theta s angle and its value during sin, cos and tan. The table consists of Sin θ, Cos θ, and tan θ for 42 values of θ. Range of -360º ≤ θ ≤ 360º. By the table we can see the range of sin, cos, and tan- -1≤Sin θ≤1, -1≤Cos θ≤1 Tan θ doesn’t have definite range.

Table of values of θ within the range of -360°≤θ≤360°

The number with E is error made by MS Excel calculations.

Highlighted means 0 and number with E will be just defined as undefined values.

This later was verified by TI-83 plus calculator.

Once again, upon the analysis of the above table of values, the values of sinθ are found to have a specific range; no smaller than -1 and no bigger than 1 (-1≤sinθ≤1). The values of cosθ have a specific range also; no smaller than -1 and no bigger than 1 (-1≤cosθ≤1). On the other hand, the values of tanθ don’t have definite range.

Four Quadrants-

Sine of theta in counter clockwise form

The value of Sin θ of range of 0<θ<90 quadrant 1, 90< θ<180 quadrant 2 are positive and 180< θ<270 quadrant 3, 270< θ<360 quadrant 4 are negative.

Sine of theta in clockwise form

The value of Sin θ of range of 0<θ<-90 quadrant 4, -90< θ<-180 quadrant 3 are negative and -180< θ<-270 quadrant 2, -270< θ<-360 quadrant 1 are positive.

Cosine of theta in counter-clockwise form

The value of Cosine of quadrant 1 0<θ<90, quadrant 4 270<θ<360 are positive and quadrant 2 90< θ<180, quadrant 3 which is 180< θ<270 are negative.

Cosine of theta in clockwise

The value of Cosine of quadrant 4 0<θ<-90, quadrant 1 -270<θ<-360 are positive and quadrant 3 -90< θ<-180, quadrant 2 which is -180< θ<-270 are negative.

Tangent of theta in counter clockwise form

The value of Tangent θ of quadrant 1 0<θ<90 and quadrant 3 which is 180< θ<270 are positive range and quadrant 2 90< θ<180, quadrant 4 270<θ<360 are negative range.

Tangent of theta in clockwise form

The value of Tangent θ of quadrant 4 0<θ<-90 and quadrant 2 which is -180< θ<-270 are negative range and quadran3 -90< θ<-180, quadrant1 -270<θ<-360 are positive range.

Axes:

The positive x axis of -360, 0, 360 degree the value of Cos θ=1, Sin θ=0, and tan θ=0.

The positive y axis of -270 and 90 degree the value of sin θ=1, Cos θ=0 and tan θ is undefined.

The negative x axis of -180 and 180 degree the value of cos θ=-1, sin θ=0, and tan θ=0.

The negative y axis of -90, 270 degree the value of sin θ =-1, cos θ=0 and tan=undefined.

To prove my angles, each sign in each quadrants, I would put any random values for theta for each right values of quadrants due to their ranges in counter clockwise and clockwise.

Counter clockwise

Just for trial, we will put a random angle for quadrant 1 which the range is 0<θ<90, in there we would put any number between 0 and 90 degrees. Such as 52, to verify the conjecture, the value of sin, cos and tan turned out to be positive. We have to remember that before using the TI-83 calculator, we should turn the mode to degree mode.

Now for quadrant 2, when we put a random angle from quadrant 2, the range of 90<θ<180, 174° in able to verify for the conjecture, the value of sin turn out to be positive while the values of cos and tan turn out to be negative.

Next, when we put a random angle from quadrant 4, the range of 180<θ<270, I would put 196 degree, 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 270<θ<360, 336° in trial to verify the conjecture, the value of cos turn out to be positive while the values of sin and tan turn out to be negative.