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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
sin
0 rad
0
0
cos
1
3 2
tan
0
3 3
Sin (-θ) = -sin θ
Cos (-θ) = cosθ
Cos (θ + π) = -cos θ
6 rad
30
4 rad
45
3 rad
60
1 2
2 2
3 2
2 2
1
1 2
3
2 rad
90
1
0
undefined
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.
2.
3.
4.
5.
6.
7.
Cos2 θ + sin2 θ = 1
Cos2 θ = ½[1 + cos(2θ)]
Sin2 θ = ½[1 – cos(2θ)]
Sin(2θ) = 2sin θ cos θ
Cos(2θ) = cos2 θ - sin2 θ
Sin(α + β) = sin α cos β + cos α sin β
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, logbx is the exponent you put on base b to get x. 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
Logb(a/c) = logba - logbc
or
Logbar = rlogba
or
bmbn = bm+n
bm/bn = bm-n
(bm)n = bmn
Graphs of Logarithmic and Exponential Functions
Notice
that
each
curve is
the
reflecti
on 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) =
than xn)
3. for |x| << l, limn
for all positive integers n (ex grows faster
(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.
1.
f( x) = 4cos (x/3)
Centerline ______________
Amplitude ______________
Period ______________
Increment ______________
2.
f ( x) = - 2 tan(x/3)
Period ______________
Vertical Asymptotes _________
3
f (x ) = (1/2) csc (x/3)
4. )
F( x) = (1/2)cot (x/2 + 𝜋/6)
Period ______________
Vertical Asymptotes _________
5.
Y = 4sin(x/2 – π/2)
Centerline ______________
Amplitude ______________
Phase Shift ______________
Period ______________
Increment ______________
6.
y = (1/2) sec (2x + π/4)
7. y = -3sec x
8.
y = 1 = 4sin(x/3)
Centerline ______________
Amplitude ______________
Phase Shift ______________
Period ______________
Increment ______________
9.
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.