Graphs of trigonometric functions
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Graphs of Trigonometric Functions. x. 0. cos x. 1. 0. -1. 0. 1. y = cos x. y. x. Graph of the Cosine Function. Cosine Function. To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts.

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Cosine function

x

0

cos x

1

0

-1

0

1

y = cos x

y

x

Graph of the Cosine Function

Cosine Function

To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.


Sine function

x

0

sin x

0

1

0

-1

0

y = sin x

y

x

Graph of the Sine Function

Sine Function

To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.


Properties of sine and cosine functions

2. The range is the set of y values such that .

5. Each function cycles through all the values of the range over an x-interval of .

Properties of Sine and Cosine Functions

Properties of Sine and Cosine Functions

The graphs of y = sin x and y = cos x have similar properties:

1. The domain is the set of real numbers.

3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.

6. The cycle repeats itself indefinitely in both directions of thex-axis.


Key steps in graphing sine and cosine
Key Steps in Graphing Sine and Cosine

  • Identify the key points of your basic graph

  • Find the new period (2π/b)

  • Find the new beginning (bx - c = 0)

  • Find the new end (bx - c = 2π)

  • Divide the new period into 4 equal parts to create new interval for the x values in key points

  • Adjust the y values of the key points by applying a and the vertical shift (d)


Secant function

The graph y = sec x, use the identity .

y

Properties of y = sec x

1. domain : all real x

x

4. vertical asymptotes:

Graph of the Secant Function

Secant Function

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

2. range: (–,–1]  [1, +)

3. period: 2


Cosecant function

To graph y = csc x, use the identity .

y

Properties of y = csc x

1. domain : all real x

x

4. vertical asymptotes:

Graph of the Cosecant Function

Cosecant Function

At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.

2. range: (–,–1]  [1, +)

3. period: 2

where sine is zero.


Key steps in graphing secant and cosecant
Key Steps in Graphing Secant and Cosecant

  • Identify the key points of your reciprocal graph (sine/cosine), note the original zeros, maximums and minimums

  • Find the new period (2π/b)

  • Find the new beginning (bx - c = 0)

  • Find the new end (bx - c = 2π)

  • Find the new interval (new period / 4) to divide the new reference period into 4 equal parts to create new x values for the key points

  • Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d)

  • Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums…

  • Graph key points and connect the dots based upon known shape


Graphs of trigonometric functions

To graph y = tan x, use the identity .

y

Properties of y = tan x

1. domain : all real x

x

4. vertical asymptotes:

period:

Graph of the Tangent Function

At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

2. range: (–, +)

3. period: 


Cotangent function

y

To graph y = cot x, use the identity .

Properties of y = cot x

x

1. domain : all real x

4. vertical asymptotes:

vertical asymptotes

Graph of the Cotangent Function

Cotangent Function

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

2. range: (–, +)

3. period: 


Key steps in graphing tan and cot
Key Steps in Graphing Tan and Cot

Identify the key points of your basic graph

  • Find the new period (π/b)

  • Find the new beginning (bx - c = 0)

  • Find the new end (bx - c = π)

  • Find the new interval (new period / 2) to divide the new reference period into 2 equal parts to create new x values for the key points

  • Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d)

  • Graph key points and connect the dots



1 chose the correct equation of the graph a y sin x b y cos x c y sin x d y cos x

y

x

1.Chose the correct equation of the graph: a) y = sin x b) y = cos x c) y = –sin x d) y = –cos x


2 chose the correct equation of the graph a y sin x b y cos x c y sin x d y cos x

y

x

2.Chose the correct equation of the graph: a) y = sin x b) y = cos x c) y = –sin xd) y = –cos x


3 chose the correct equation of the graph a y sin x b y 2 sin x c y sin 2x d y sin x

y

x

3. Chose the correct equation of the graph: a) y = sin ½ x b) y = 2 sin x c) y = sin 2x d) y = sin x


4 chose the correct equation of the graph a y sin x 1 b y cos x 1 c y 2sin x d y 2cos x

y

x

4. Chose the correct equation of the graph: a) y = sin x + 1 b) y = cos x + 1 c) y = 2sin x d) y = 2cos x


Graphs of trigonometric functions

y

x

5. Chose the correct equation of the graph: a) y = sin x + 1 b) y = sin (x + π/2)c) y = sin (x – π/2)d) y = sin x


6 chose the correct equation of the graph a y sin 4x b y 4 sin x c y sin x 4 d y sin x 4

y

x

6. Chose the correct equation of the graph: a) y = sin 4x b) y = 4 sin x c) y = sin x + 4 d) y = sin (x + 4)


7 chose the correct equation of the graph a y sin x b y 2 sin x c y sin 2x d y sin x

y

x

7. Chose the correct equation of the graph: a) y = sin ½ x b) y = 2 sin x c) y = sin 2x d) y = sin x


8 chose the correct equation of the graph a y 4sin x b y 3 sin x 1 c y sin 3x 1 d y sin x 4

y

x

8. Chose the correct equation of the graph: a) y = 4sin x b) y = 3 sin x + 1 c) y = sin 3x + 1 d) y = sin x + 4


Secant function1

y

x

Secant Function

9. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x


Secant function2

y

x

Secant Function

10. Chose the correct equation of the graph: a) y = sec x

b) y = 2sec x c) y = sec 2x

d) y = sec x + 2


Secant function3

y

x

Secant Function

11. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x


Cosecant function1

y

x

12. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x

.

Cosecant Function


Cosecant function2

y

x

13. Chose the correct equation of the graph: a) y = csc x

b) y = csc 2x c) y = 2csc x

d) y = csc x – 2

.

Cosecant Function


Cotangent function1

y

x

14. Chose the correct equation of the graph: a) y = tan x

b) y = cot x c) y = -tan x

d) y = -cot x

.

Cotangent Function


Graphs of trigonometric functions

y

x

period:

15. Chose the correct equation of the graph: a) y = tan x

b) y = cot x c) y = -tan x

d) y = -cot x

.


Graphs of trigonometric functions

y

period:

x

16. Chose the correct equation of the graph: a) y = 2tan x

b) y = tan 2x c) y = -2cot x

d) y = -cot 2x

.