University Mathematics Handbook (2015)

II. Functions

Chapter 2. Classes of Elementary Functions

2.1  Linear Function

2.2  Square Function

2.3  Power Function ,

Examples are in Figures 1-4.

Figure 1      Figure 2

Figure 3      Figure 4

2.4  Exponential Function

(Figures 5, 6)

Figure 5      Figure 6

2.5  Logarithm Function

. Graph is shown in Figure 7.

If its basis is , it's denoted .

If its basis is , it's denoted

Function is the inverse of . Their graphs are symmetric with respect to the straight line (Figure 8).

Figure 7      Figure 8

Basic formulas:

2.6  Trigonometric Functions

a.  Sine

is an odd periodic function with a period (Figure 9).

Figure 9

b.  Cosine

is an even periodic function with a period.

: Its graph results from shifting the graph by leftward (Figure 10).

Figure 10

c.  Tangent

is an odd periodic function with a period (Figure 11).

d.  Cotangent

is an odd periodic function with a period (Figure 12).

    Figure 11      Figure 12

e.  Increase and Decrease Domains and Denotations of Trigonometric Functions

f.  Basic Formulas

1.  

2.  

3.  

4.  

5.  

6.  

7.  

8.  

9.  

10.  

11.  

12.  

13.  

14.  

g.  Reduction Formulas

h.  Additional Formulas

1.  , where is satisfy the equations , and

2.  

3.  

4.  

5.  

i.  Expressing Trigonometric Function Through Another

In these formulas, the root's mark is the same as that of the left side of the function, which depends on the quarter being .

Example: If , then when is computed through , it is taken with , since in quarter II function is positive, that is, . Still, for the same angle, is taken with , since, in the second quarter, , and therefore, .

j.  Law of Sines

For a triangle of sides, and angles, respectively

where is the radius of the circumscribed circle.

k.  Law of Cosines

The square of one triangle side equals the sum of the squares of the other two sides minus twice the product of these two sides and the cosine of the angle between them:

.

2.7  Inverse Trigonometric Functions

Trigonometric functions are periodical functions, and therefore not bijective. Reducing their natural domain, we get invertible functions:

a.  The Inverse function of is

(arcsine x).

The graph of function is symmetric to the graph of with respect to the straight line (Figure 13).

is an odd function.

b.  The inverse of function is (arccosine ) (Figure 14).

Figure 13      Figure 14

c.  The inverse function of is (arctangent ).

is an odd function (Figure 15).

d.  The inverse function of is (arccotangent )

(Figure 16).

    Figure 15       Figure 16

e.  Additional formulas:

1. 

2. 

3. 

4. 

5. 

6. 

2.8  Hyperbolic Functions

a.  Hyperbolic sine: is an odd function (Figure 17).

b.  Hyperbolic cosine: is an even function (Figure 17).

c.  Hyperbolic tangent: is an odd function (Figure 18).

d.  Hyperbolic cotangent: is an odd function (Figure 18).

    Figure 17    ‘    Figure 18

2.9  Basic Formulas

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. 

11.