Lesson 5.1 Graphing sine and cosine function

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(i) Graph sine, cosine and tangent function

Characteristics:« domain and range« amplitude« period« zeros

(ii) Transformations of the sine and cosine function

(iii) Solve first and second degree trigonometric equations algebraically and graphically.

CHAPTER 5

Trigonometric Functions and Graphs

Lesson 5.1 Graphing sine and cosine function

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Special Angles on the Unit Circle

Lesson 5.1 Graphing sine and cosine function

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Summary:

Lesson 5.1 Graphing sine and cosine function

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Lesson 5.1 Sketch the graph of and .

Choose 5 key points associated with each graph

determine the characteristics of the graph

A. Graph the function

The sine function

y­coordinate of the point

Lesson 5.1 Graphing sine and cosine function

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Characteristics:

periodic function: a function that repeats itself over regular intervals (cycles) of its domain.

(i) period: the portion of the graph which illustrates one full cycle of the function.

one cycle:

sinusoidal function: a periodic function that looks like waves; oscillates repeatedly up and down from a centre line

Lesson 5.1 Graphing sine and cosine function

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(ii) Maximum/Minimum

(iii) Domain/Range:

(iv) Horizontal Central Axis : the horizontal line which is halfway between the local maximum and local minimum.

Characteristics cont'd:

Equation of the Sinusoidal Axis

Midline

Lesson 5.1 Graphing sine and cosine function

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Characteristics cont'd:

(v) Amplitude: vertical distance from the sinusoidal axis to a local maximum or minimum

(vi) x and y­intercepts

Lesson 5.1 Graphing sine and cosine function

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Choose 5 key points associated with each graph

determine the characteristics of the graph

B. Graph the function

The cosine function

the x­coordinate of the point

Lesson 5.1 Graphing sine and cosine function

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(i) Maximum/Minimum:

(ii) Domain/Range:

(iv) Amplitude:

(iii) Horizontal Central Axis: (Equation of the Sinusoidal Axis)

(v) x and y­intercepts

Characteristics:

one cycle:

Lesson 5.1 Graphing sine and cosine function

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Compare the graph of to .

Lesson 5.1 Graphing sine and cosine function

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Transformations: Vertical and Horizontal Stretches

Example1

Identify the amplitude for each function.

Vertical Stretch

(i)

affects the ______________

Lesson 5.1 Graphing sine and cosine function

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Example 1 cont'd

(ii)

(iii)

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Horizontal Stretch

Example 2

Identify the horizontal stretch and the period for each function.

period: period:

b= b=

Solve an inequality to determine the length of the cycle and the key points (max,min,x­ints, hits SA)

affects the ______________

Lesson 5.1 Graphing sine and cosine function

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Example 3 Graph the functions What is the period of ?

Solve an inequality

period:

b=

period:

b=

Lesson 5.1 Graphing sine and cosine function

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What is the period of ?

Solve an inequality

Example 4: Graph the functions

period:

b=period:

b=

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Example 6: What is the period of ?

Example 5: What is the period of ?

Lesson 5.1 Graphing sine and cosine function

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Example 7

Sketch the graph of for at least one cycle.

Determine • the amplitude • period • maximum and minimum values• x and y­intercepts • domain and range.

Method 1: Use Mapping Rule

Lesson 5.1 Graphing sine and cosine function

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Method 2: Use Inequality and Key Points

P.233­236 #4­5 (no graph), 6­10, 14