Mathematics Dictionary

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 1 in the lower box. We can see the curve of y = sin(x).
- To see y = cos(x), Click Back and click 2 in lower box

Go to Begin

TR 06 02. Graph of y = cos(x)

Properties

1. Range is between -1 and 1.
2. Period is 2*pi.
3. Zeros of y at x = n*pi + pi/2.
4. Values of y is +1 if x = 2*n*pi
5. Values of y is -1 if x = (2*n+1)*pi
6. 1st derivative is y' = -sin(x).
7. 2nd derivative is y" = -cos(x).

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 2 in the lower box. We can see the curve of y = cos(x).
- To see y = sin(x), Click Back and click 1 in lower box

Go to Begin

TR 06 03. Graph of y = tan(x)

Properties

1. Range is between -∞ and +∞.
2. Period is pi.
3. Zeros of y at x = n*pi.
4. Values of y is -∞ and +∞ if x = n*pi + pi/2
5. Asymptotes at x = n*pi + pi/2.
6. 1st derivative is y' = +sec(x)^2.
7. 2nd derivative is y" = ?.

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 3 in the lower box. We can see the curve of y = tan(x).

Asymptote of y = tan(x)

Since x = pi/2, x = 3*pi/2, x = 3*pi/2, .... and y = infinite.
Hence x = pi/2, x = 3*pi/2, x = 3*pi/2, .... are asymptote of y = tan(x)
General solution is x = n*pi + pi/2.

Go to Begin

TR 06 04. Graph of y = csc(x)

Properties

1. Range is between -1 and -∞ and 1 and +∞ or y GE Abs(1).
2. Period is 2*pi.
3. Zeros of y : none.
4. Values of y is between -1 and -∞ and between 1 and +∞.
5. Asymptotes at x = n*pi.
6. 1st derivative is y' = ?.
7. 2nd derivative is y" = ?.

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 4 in the lower box. We can see the curve of y = csc(x).

Asymptote of y = csc(x)

Since x = 0, x = pi, x = 2*pi, .... and y = infinite.
Hence x = 0, x = pi, x = 2*pi, .... are asymptote of y = csc(x)
General solution is x = n*pi.

Go to Begin

TR 06 05. Graph of y = sec(x)
Properties

1. Range is between -∞ and +∞.
2. Period is pi.
3. Zeros of y : x = pi/2, 3*pi/2, ....
4. Values of y is between -∞ and +∞.
5. Asymptotes at x = n*pi.
6. 1st derivative is y' = ?.
7. 2nd derivative is y" = ?.

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 5 in the lower box. We can see the curve of y = sec(x).

Asymptote of y = sec(x)

Since x = 0, pi, x = 2*pi, .... and y = infinite.
Hence x = 0, pi, x = 2*pi, .... are asymptote of y = sec(x)
General solution is x = n*pi.

Go to Begin

TR 06 06. Graph of y = cot(x)
Properties

1. Range is between -1 and -∞ and 1 and +∞ or y GE Abs(1).
2. Period is 2*pi.
3. Zeros of y : none.
4. Values of y is between -1 and -∞ and between 1 and +∞.
5. Asymptotes at x = n*pi.
6. 1st derivative is y' = ?.
7. 2nd derivative is y" = ?.

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 6 in the lower box. We can see the curve of y = cot(x).

Asymptote of y = cot(x)

Since x = 0, x = pi, x = 2*pi, .... and y = infinite.
Hence x = 0, x = pi, x = 2*pi, .... are asymptote of y = cot(x)
General solution is x = n*pi.

Go to Begin

TR 06 07. Properties of functions

Properties

1. Peridical function : If F(x+p) = F(x), then F(x) is periodic with period = p.
- y = sin(x) and period is 2*pi.
- y = sin(2*x) and period is 2*pi/2.
- y = sin(4*x) and period is 4*pi/2.
2. Even function : If F(-x) = F(x), then F(x) is even functions.
- y = cos(x) is even function.
- y = sec(x) is even function.
3. Odd function : If F(-x) = -F(x), then F(x) is odd functions.
- y = cos(x) is even function.
- y = sec(x) is even function.

Functions

Since sin(-x) = -sin(x), hence y = sin(x) is odd function.
Since tan(-x) = -tan(x), hence y = tan(x) is odd function.
Since cos(-x) = +cos(x), hence y = cos(x) is even function.

Graphic shapes

Based on 1st derivative
- If y' is less than 0, the curve is decreasing.
- If y' is greater than 0, the curve is increasing.
- if y' is equal to 0, the curve has extreme point.
Based on 2nd derivative
- If y" is less than zero, the curve is concave downward.
- If y" is greater than zero, the curve is concave upward.
- If y" is equal to zero, the curve has point of inflexion.

Asymptotes

If x = a and F(x) goes to infinite, then x = a is vertical asymptote for y = F(x).
If x = infinite and F(x) = a, then y = a is horizontal asymptote for y = F(x).

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TR 06 08. Between 0 and 2*pi and sin(x) GT cos(x), find x

Values

.... x 0 ..... 45 .. 90 ..... 135 ... 180 ..... 225 .. 270 ..... 315 .. 360
cos(x) 1 Sqr(2)/2 ... 0 -Sqr(2)/2 .... -1 -Sqr(2)/2 .... 0 +Sqr(2)/2 .... 1
sin(x) 0 Sqr(2)/2 ... 1 -Sqr(2)/2 ..... 0 -Sqr(2)/2 ... -1 -Sqr(2)/2 .... 0

Solution

From above table we see that
if sin(x) GT cos(x) the x values are between 45 and 225 degrees

Go to Begin

TR 06 09. Sketch y = cos(x) + sin(x) + Abs(cos(x) - sin(x))
Solurion : See TR 27 09

x between 0 and pi/4 : y = 2*cos(x)
x between pi/4 and 5*pi/4 : y = 2*sin(x)
x between 5*pi/4 and 2*pi : y = 2*cos(x)

Diagrams

Subject | Diagrams of trigonometric functions
Click subject to enter program GC.
To see the graph
- Click start to get menu
- Click topic 4 in the upper box.
- Click 7 in the lower box. We can see the curve

Go to Begin

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