Review of Trigonometric, Logarithmic, and Exponential Functions

In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications.

Trigonometric Functions

Geometrically, there are two ways to describe trigonometric functions:

Polar Angle

x = cosθ

y = sinθ

Measure θ (in radians):

θ = arc length / radius

1 revolution = 2πr / r = 2π

1° = (1 revolution) / 360 = 2π / 360 = π/180

Right Triangle

sinθ = opposite / hypotenuse = y/r

cosθ = adjacent / hypotenuse = x/r

tanθ = opposite / adjacent = y/x

cscθ = 1/( sinθ) = r/y

secθ = 1/( cosθ) = r/x

cotθ = 1/( tanθ) = x/y

Evaluating Trigonometric Functions

0  rad
       π/6  rad
       π/4  rad
       π/3 rad
       π/2  rad
0°
       30°
       45°
       60°
       90°
sinθ
0
       1/2
       √2/2
       √3/2
       1
cosθ
1
       √3/2
       √2/2
       1/2
       0
tanθ
0
       √3/3
       1
       √3
       undefined

sin(-θ) = -sinθ
             sin(θ+π/2) = cosθ
cos(-θ) = cosθ
             cos(θ+π/2) = -sinθ
cos(θ+π) = -cosθ
             cos(θ+2π) = cosθ
sin(θ+π) = -sinθ
             sin(θ+2π) = sinθ

Trigonometric Identities

We list here some of the most commonly used identities:

1.  cos2θ+sin2θ = 1
             6.   sin(α+β) = sinαcosβ+cosαsinβ
2.  cos2θ = 1
2
 [1+cos(2θ)]
             7.   cos(α+β) = cosαcosβ-sinαsinβ
3.   sin2θ = 1
2
 [1-cos(2θ)]            
4.   sin(2θ) = 2sinθcosθ
             8.  C1cos(ωx)+C2sin(ωx) = Asin(ωx+φ)
5.  cos(2θ) = cos2θ-sin2θ
                   where A =   _______
√C12+C22
 
 ,    φ = arctan(C1/C2)

Graphs of Trigonometric Functions
sin (x) cos (x)
tan (x) csc (x)
sec (x) cot (x)

 
 

Logarithmic and Exponential Functions

Logarithmic and exponential functions are inverses of each other:

y = logb x       if and only if       x = by
y = ln x       if and only if       x = ey
In words, logb x is the exponent you put on base b to get x. Thus,
logb bx = x        and        blogb x = x.

More Properties of Logarithmic and Exponential Functions

Notice the relationship between each pair of identities:

logb 1 = 0
↔
b0 = 1
       logb ac = logb a+logb c
↔
bmbn = bm+n
logb b = 1
↔
b1 = b
       logb a
c
 = logb a-logb c
↔
bm
bn
 = bm-n
logb 1
c
 = -logb c
↔
b-m = 1
bm
       logb ar = rlogb a
↔
(bm)n = bmn

Graphs of Logarithmic and Exponential Functions
f(x)=ln xf(x)=ex
Notice that each curve is the reflection of the other about the line y = x

 
 

Limits of Logarithmic and Exponential Functions

  1. limx→∞ [ln x / x] = 0    (ln x grows more slowly than x).

  2. limx→∞ [(ex)/(xn)] = ∞ for all positive integers n    ( ex grows faster than xn).

  3. limn → ∞(1 + x/n)n = ex  for all real number x

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