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

Envision the Unit Circle – now draw your version:

x = cos θ

y = sin θ

measure θ in radians:

θ = arc length/ radius

For example, 180o = πr/r = π radians

Radians = [degrees/180] π

Right Triangle

Draw a right triangle with base angle θ, opposite side y, adjacent side x, and hypotenuse r

Sin θ = opposite/hypotenuse = y/r

Cos θ = adjacent/hypotenuse = x/r

Tan θ = opposite/adjacent = y/x

Csc θ = 1/sin θ = r/y

Sec θ = 1/cos θ = r/x

Cot θ = 1/tan θ = x/y

Evaluating Trigonometric Functions

0rad / 6rad / 4rad / 3rad / 2rad
0 / 30 / 45 / 60 / 90
sin / 0 / 12 / 22 / 32 / 1
cos / 1 / 32 / 22 / 12 / 0
tan / 0 / 33 / 1 / 3 / undefined

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

Trigonometric Identities

We list here some of the most commonly used identities:

1. Cos2 θ + sin2 θ = 1
2. Cos2 θ = ½[1 + cos(2θ)]
3. Sin2 θ = ½[1 – cos(2θ)]
4. Sin(2θ) = 2sin θ cos θ
5. Cos(2θ) = cos2 θ - sin2 θ
6. Sin(α + β) = sin α cos β + cos α sin β
7. Cos(α + β) = cos α cos β - sin α sin β

Graphs of Trigonometric Functions

sinx / cosx
tanx / cotx
secx / cscx

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,logbxis the exponent you put on basebto getx. Thus,

Logb bx = x and blogbx = x

More Properties of Logarithmic and Exponential Functions

Notice the relationship between each pair of identities:

Logb 1 = 0 or b0 = 1

Logb b = 1 or b1 = b

Logb (1/c) = -logb c or b-m = 1/bm

Logbac = logb a + logb c or bmbn = bm+n

Logb(a/c) = logba - logbc or bm/bn = bm-n

Logbar = rlogba or (bm)n = bmn

Graphs of Logarithmic and Exponential Functions

/ Notice that each curve is the reflection of the other about the liney=x. /
f(x)=lnx / f(x)=ex

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. for |x| < l, limn (1 + (x/n))n = ex.

NOW FOR THE PROBLEMS THAT WILL HELP PREPARE YOU FOR AP PHYSICS C

Sketch a graph of the function and fill in the blanks. Include two full periods.

f( x) = 4cos (x/3)

Centerline ______

Amplitude ______

Period ______

Increment ______

f ( x) = - 2 tan(x/3)

Period ______

Vertical Asymptotes ______

f (x ) = (1/2) csc (x/3)

4. )

F( x) = (1/2)cot (x/2 + π/6)

Period ______

Vertical Asymptotes ______

Y = 4sin(x/2 – π/2)

Centerline ______

Amplitude ______

Phase Shift ______

Period ______

Increment ______

y = (1/2) sec (2x + π/4)

7. y = -3sec x

y = 1 = 4sin(x/3)

Centerline ______

Amplitude ______

Phase Shift ______

Period ______

Increment ______

y = 1 – 2csc2x

10.

y = 4cos(4x + π/2)

Centerline ______

Amplitude ______

Phase Shift ______

Period ______

Increment ______

11.

Chemco Manufacturing estimates that its profit P in hundreds of dollars is

P = -2x2 + 16x + 2 , where x is the number of units produced in thousands. How many units must be produced to obtain a maximum profit?

A. 4 units B. 32 units C. 3200 units D. 4000 units.

12.

Find the height of a tree on a hillside of slope 32o (from the horizontal). At a point

75 feet down the hill from the tree, the angle of elevation to the top of the tree is 48o.

13.

To approximate the length of a marsh, a surveyor walks 450 meters from point A to point B, turns 65o and walks 325 meters to point C. What is the length of the

marsh(AC)?

14.

Solve the system of equations:

7x + 12y = 63

2x + 3y = 15

15.

A contractor is hiring two trucking companies to haul 1600 tons of crushed stone for a highway construction project. The contract states that company A must haul 4 times as herr17

much stone as the company B. Find the amount of stone hauled by each company.