Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 6: The Trigonometric Functions And Their Inverses – Section 6.1.5: Graphs Of Trigonometric Functions

 

6.1.5
Graphs Of Trigonometric Functions

 

 

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1. Note

 

 

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2. The Graph Of Sine

 

The unit circle is drawn in Fig. 2.2. The graph of the sine function y = sin x is sketched in Fig. 2.3.

 

 

 

Symmetry. Since sin x) = – sin x, sin x is an odd function, so its graph is symmetric with respect to the origin.

 

 

Using The First And Second Derivatives. Based on Section 5.6, we can use the first and second derivatives of sin x to get

 

Fig. 2.1

 

A Chart For y = sin x.

 

 

Fig. 2.2

 

 

Fig. 2.3

 

Graph Of y = sin x.

 

Example 2.1

 

Sketch the graph of y = 2 sin x.

 

Solution

 

Fig. 2.4

 

 

Fig. 2.5

 

Graph Of
y = 2 sin x.

EOS

 

Note that to find the period p > 0 of a function f(x) we try to establish the relation f(x) = f(x + p) that's true for all x in
dom( f ). This relation is in reverse order to the relation f(x + p) = f(x) in the definition.

 

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3. The Graph Of Cosine

 

 

Symmetry. cosx) = cos x for all x; cos x is an even function; its graph is symmetric with respect to the y-axis.

 

 

Fig. 3.1

 

 

Fig. 3.2

 

Graph Of y = cos x.

 

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4. The Graph Of Tangent

 

The unit circle is drawn in Fig. 4.2. The graph of the tangent function y = tan x is sketched in Fig. 4.3.

 

 

 

Fig. 4.1

 

 

 

Fig. 4.2

 

 

 

Fig. 4.3

 

Graph Of y = tan x.

 

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5. The Graph Of Cotangent

 

 

Fig. 5.1

 

 

Fig. 5.2

 

Graph Of y = cot x.

 

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6. The Graph Of Secant

 

The unit circle is drawn in Fig. 6.2. The graph of the secant function y = sec x is sketched in Fig. 6.3.

 

 

 

Fig. 6.1

 

 

The symbol “~” means “ has
the same sign as”.

 

 

Fig. 6.2

 

 

Fig.  6.3

 

Graph Of y = sec x.

 

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7. The Graph Of Cosecant

 

 

Fig. 7.1

 

 

Fig. 7.2

 

Graph Of y = csc x.

 

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Problems & Solutions

 

 

 

 

 

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4.  Sketch the graph of y = tan2 x.

 

Solution

 

 

 

 

 

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In the following chart, the symbol “~” means “ has the same sign as”.

 

 

 

 

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