Mathematics Dictionary
Dr. K. G. Shih
Trigonometric functions in rectangular coordinates
Subjects
Symbol Defintion
Example Sqr(x) = Square root of x
AN 03 00 |
- Outlines
AN 03 01 |
- Graph of y = sin(x)
AN 03 02 |
- Graph of y = cos(x)
AN 03 03 |
- Graph of y = tan(x)
AN 03 04 |
- Graph of y = csc(x)
AN 03 05 |
- Graph of y = sec(x)
AN 03 06 |
- Graph of y = tan(x)
AN 03 07 |
- Quick sketch of y = sin(x)
AN 03 08 |
- Quick sketch of y = tan(x)
AN 03 09 |
- Quick sketch of y = sec(x)
AN 03 10 |
- Properties of graph of y = sin(x)
AN 03 11 |
- Properties of graph of y = tan(x)
AN 03 12 |
- Properties of graph of y = sec(x)
AN 03 10 |
- Questions
AN 03 14 |
- Reference
Answers
AN 03 01. Graph of y = sin(x)
Defintion of sine functions
Sin(A) = Opp/Hyp in right trangle ABC with C = 90 degrees
Sin(A) = y/r in rectangular coordiantes r = Sqr(x^2 + y^2)
Properties of y = sin(x)
Period is 2*pi
Range is between -1 and 1
Zeros at x = 0, 180, 360, ... or x = n*pi
Maximum value = +1 at x = 090, 450, .... or x = 2*n*pi + pi/2
Minimum value = -1 at x = 270, 630, .... or x = (2*n+1)*pi + pi/2
Diagram
Graphs in Analytic geometry
Section 3 in anlytic geometry
Go to Begin
AN 03 02. Graph of y = cos(x)
Defintion of cosine functions
Cos(A) = Adj/Hyp in right trangle ABC with C = 90 degrees
Cos(A) = x/r in rectangular coordiantes with r = Sqr(x^2 + y^2)
Properties of y = cos(x)
Period is 2*pi
Range is between -1 and 1
Zeros at x = 090, 270, ... or x = (2*n+1)*pi + pi/2
Maximum value = +1 at x = 000, 360, .... or x = 2*n*pi
Minimum value = -1 at x = 180, 540, .... or x = (2*n+1)*pi
Diagram
Graphs in Analytic geometry
Section AN 03
Go to Begin
AN 03 03. Graph of y = tan(x)
Defintion of tangent functions
Tan(A) = Opp/Adj in right trangle ABC with C = 90 degrees
Tan(A) = y/x in rectangular coordiantes
Properties of y = tan(x)
Period is pi
Range is between -infinite and +infinite
Zeros at x = 0, 180, 360
Value = +1 at x = 045, 225
Value = -1 at x = 135, 315
Asymptotes at x = 090, 270
Diagram
Graphs in Analytic geometry
Section 3 in anlytic geometry
Go to Begin
AN 03 04. Graph of y = csc(x)
Defintion of cosine functions
Csc(A) = Hyp/Opp in right trangle ABC with C = 90 degrees
Csc(A) = r/y in rectangular coordiantes with r = Sqr(x^2 + y^2)
Properties of y = csc(x)
Period is 2*pi
Range is Abs(y) > 1
Zeros of y : none
Maximum point = -1 at x = 270
Minimum point = +1 at x = 0, 630
Asymptotes : x = 0 or x = pi or x = 2*pi
Diagram
Graphs in Analytic geometry
Section AN 03
Go to Begin
AN 03 05. Graph of y = sec(x)
Defintion of cosine functions
Sec(A) = Hyp/Adj in right trangle ABC with C = 90 degrees
Sec(A) = r/x in rectangular coordiantes with r = Sqr(x^2 + y^2)
Properties of y = sec(x)
Period is 2*pi
Range is Abs(y) > 1
Zeros of y : none
Maximum point = -1 at x = 0, 360
Minimum point = +1 at x = 270
Asymptotes : x = 90 or x = 270 degrees
Diagram
Graphs in Analytic geometry
Section AN 03
Go to Begin
AN 03 06. Graph of y = cot(x)
Defintion of tangent functions
Tan(A) = Adj/Opp in right trangle ABC with C = 90 degrees
Tan(A) = x/y in rectangular coordiantes
Properties of y = tan(x)
Period is pi
Range is between -infinite and +infinite
Zeros at x = 090, 270
Value = +1 at x = 045
Value = -1 at x = 135
Asymptotes at x = 0, 180, 360
Diagram
Graphs in Analytic geometry
Section 3 in anlytic geometry
Go to Begin
AN 03 07. Quick sketch y = sin(x)
Use five special points
Point 1 : x = 000 and y = sin(000) = +0
Point 2 : x = 090 and y = sin(090) = +1
Point 3 : x = 180 and y = sin(180) = +0
Point 4 : x = 270 and y = sin(270) = -1
Point 5 : x = 360 and y = sin(360) = +0
Sketch
Put above 5 points on oxy coordiante system
Draw a smooth curve using thes 5 points
Demo diagram
Graphs in Analytic geometry
Section AN 21 Program 03 07
Go to Begin
AN 03 08. Quick sketch y = tan(x)
Use three special points between 0 and 360 degrees
Point 1 : x = 000 and y = tan(000) = +0
Point 2 : x = 180 and y = tan(090) = +0
Point 3 : x = 360 and y = tan(360) = +0
Use two asymptotes : tan(90) = infinite and tan(270) = infinite
Vertical asymptote at x = 090 degrees
Vertical asymptote at x = 270 degrees
Sketch
Draw two two vertical asymptotoes : x = 90 and x = 180 degrees
Put above 3 points on oxy coordiante system
Draw a smooth curve from (0,0) to infinite at x = 90
Draw a smooth curve from -infinite at x = 90 to +infinite at x x = 270
Draw a smooth curve from -infinite at x = 90 to 0 x = 360
Demo diagram
Graphs in Analytic geometry
Section AN 21 Program 03 08
Go to Begin
AN 03 09. Quick sketch y = sec(x)
Use three special points between 0 and 360 degrees
Point 1 : x = 000 and y = sec(000) = +1
Point 2 : x = 180 and y = sec(090) = -1
Point 3 : x = 360 and y = sec(360) = +1
Use two asymptotes : tan(90) = infinite and tan(270) = infinite
Vertical asymptote at x = 090 degrees
Vertical asymptote at x = 270 degrees
Sketch
Draw two two vertical asymptotoes : x = 90 and x = 180 degrees
Put above 3 points on oxy coordiante system
Draw a smooth curve from (0,0) to infinite at x = 90
Draw a smooth curve from -infinite at x = 90 to +infinite at x x = 270
Draw a smooth curve from -infinite at x = 90 to 0 x = 360
Demo diagram
Graphs in Analytic geometry
Section AN 21 Program 03 09
Answer
Go to Begin
AN 03 10. Properties of graph of y = sin(x)
Properties : x = 0 to x = 2*pi
It is a sine curve
The range is from -1 to 1
The period is 2*pi
The y-intercept is 0
The zeros of y is at x = 0, x = pi and x = 2*pi
The concavity
It is concave downward from x = 0 to x = pi
It is concave upward from x = pi to x = 2*pi
The increasing and decreasing
It is increasing from x = 0 to x = pi/2 and x = 3*pi/2 to 2*pi
It is decreasing from x = pi/2 to x = 3*pi/2
The extreme points
y = +1 when x = pi/2
y = -1 when x = 3*pi/2
Go to Begin
AN 03 11. Properties of graph of y = tan(x)
Properties : x = 0 to x = 2*pi
It is a tangent curve
The range is from -infinite to +infinite
The period is pi
The y-intercept is 0
The zeros of y is at x = 0, x = pi and x = 2*pi
The concavity
It is concave upward from x = 0 to x = pi/2
It is concave downward from x = pi/2 to x = pi
It is concave upward from x = pi to x = 3*pi/2
It is concave downward from x = 3*pi/2 to x = 2*pi
The increasing and decreasing
It is always increasing from x = 0 to 2*pi
The extreme points : None
Asymptotes
Vertical asymptote : x = pi/2 and x = 3*pi/2
Go to Begin
AN 03 12. Properties of graph of y = sec(x)
Properties : x = 0 to x = 2*pi
It is a secant curve
The range is from -infinite to -1 and from 1 to +infinite
The period is 2*pi
The y-intercept : at y = 1
The zeros of y : None (Since -1 LT y GT 1)
The concavity
It is concave upward from x = 0 to x = pi/2
It is concave downward from x = pi/2 to x = 3*p/2i
It is concave upward from x = 3*pi/2 to x = 2*pi
The increasing and decreasing
It is increasing from x = 0 to pi and from x = 3*pi/2 to 2*pi
It is decreasing from x = pi to 3*pi/2
The extreme points
Minimum at x = 0
Maximum at x = pi
Asymptotes
Vertical asymptote : x = pi/2 and x = 3*pi/2
Go to Begin
AN 03 13. Questions
Diagram
Diagrams
Section AN 21 03
Examples : Use above diagram program
1. Find period of y = sin(2*x)
2. Find period of y = sin(4*x)
3. Find period of y = tan(2*x)
4. Find period of y = tan(4*x)
Examples
Solve sin(x) = 2
Solve sec(x) = 1
Examples
Sketch y = sin(x) from x = 0 to x = 360
Sketch y = cos(x) from x = 0 to x = 360
Write the vertical asymptotes of y = tan(x) in general form
Go to Begin
AN 03 14. References
Diagram
Diagrams
Section AN 21 03
Diagrams
Section GC 04
Trigonometry
Diagrams
Text of trigonometry
Go to Begin
AN 03 00. Outlines
Period
y = sin(x), period is 2*pi
y = cos(x), period is 2*pi
y = csc(x), period is 2*pi
y = sec(x), period is 2*pi
y = tan(x), period is 1*pi
y = cot(x), period is 1*pi
Range
y = sin(x) and y = cos(x), range is from -1 to 1
y = sec(x) and y = csc(x), range is Abs(y) GT 1
y = tan(x) and y = cot(x), range is from -infinite to +infinite
Go to Begin
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