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

 1. Note 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.

 3. The Graph Of Cosine Symmetry. cos(–x) = 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.

 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.

 5. The Graph Of Cotangent  # Fig. 5.1  Fig. 5.2   Graph Of y = cot x.

 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.

 7. The Graph Of Cosecant  Fig. 7.1  Fig. 7.2   Graph Of y = csc x.

 Problems & Solutions               4.  Sketch the graph of y = tan2 x.

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