Download Algebra 2 Section 13.1 Notes Exploring Periodic Data

Algebra 2 Section 13.1 Notes
Exploring Periodic Data
A periodic function is a function that repeats a pattern of y-values (outputs) at regular
intervals. One complete pattern is a cycle. The period of a function is the horizontal
length (the distance along the x-axis) of one cycle. The x-value in a periodic
function often represents time.
The midline is the horizontal line midway between the maximum and minimum values of
a periodic function. The amplitude is half the difference between the maximum
and minimum values the function.
Algebra 2 Section 13.2 Notes
Angles and the Unit Circle
An angle in the coordinate plane is in standard position when the vertex is at the origin
and one ray is on the positive x-axis. The ray on the x-axis is the initial side of the
angle. The other ray is the terminal side of the angle.
The measure of an angle in standard position is the amount of rotation from the initial side
to the terminal side. The measure of an angle is positive when the rotation from the
initial side to the terminal side is in the counterclockwise direction. The measure is
negative when the rotation is clockwise.
Two angles in standard position are coterminal angles if they have the same terminal side.
The unit circle has a radius of 1 and its center at the origin.
The cosine of θ (cos θ) is the x-coordinate of the point at which the terminal side of the
angle intersects the unit circle. The sine of θ (sin θ) is the y-coordinate.
Algebra 2 Section 13.3 Notes
Radian Measure
A central angle of a circle is an angle with a vertex at the center of a circle. An
intercepted arc is the portion of the circle with endpoints on the sides of the central
angle and remaining points within the interior of the angle.
A radian is the measure of a central angle that intercepts an arc with length equal to the
radius of the circle. Radians, like degrees, measure the amount of rotation from the
initial side to the terminal side of an angle
An angle with a full circle rotation measures 2л radians. An angle with a semicircle
rotation measures л radians.
d
" r" radians

You can use the proportion
to convert between radians and degrees.

" " radians
180
" " radians
.
180 
180 
To convert radians to degrees, multiply by
.
" " radians
To convert degrees to radians, multiply by
For a circle of radius r and a central angle of measure θ in radians, the length s of the
intercepted arc is: s = rθ.
Algebra 2 Section 13.4 Notes
The Sine Function
The sine function, y = sinθ, matches the measure θ of an angle in standard position with
the y-coordinate of a point on the unit circle. This point is where the terminal side
of the angle intersects the unit circle.
The graph of a sine function is called a sine curve. By varying the period (horizontal
length of one cycle), you get different sine curves.
Properties of Sine Functions: Suppose y = a sin bθ , with a ≠ 0, b > 0 and θ in radians…
|a| is the amplitude of the function;
2
is the period of the function; and…
b
b is the number of cycles in the interval from 0 to 2л .
You can use 5 points equally spaced through one cycle to sketch a sine curve. For a > 0,
this 5 point pattern is zero-max-zero-min-zero.
Algebra 2 Section 13.5 Notes
The Cosine Function
The cosine function, y = cosθ, matches θ with the x-coordinate of the point on the unit
circle where the terminal side of angle θ intersects the unit circle. The symmetry of
the set of points (x, y) = (cosθ, sinθ) on the unit circle guarantees that the graphs of
sine and cosine are congruent translations of each other.
Properties of Cosine Functions: Suppose y = a cos bθ, with a ≠ 0, b > 0 and θ in radians:
|a| is the amplitude of the function;
2
is the period of the function; and…
b
b is the number of cycles in the interval from 0 to 2л .
You can use 5 points equally spaced through one cycle to sketch a cosine curve. For a > 0,
this 5 point pattern is max-zero-min-zero-max.
Algebra 2 Section 13.6 Notes
The Tangent Function
The tangent function has infinitely many points of discontinuity with a vertical asymptote
at each point. Its range is all real numbers. Its period is л, half that of both the sine
and cosine functions. Its domain is all real numbers except odd multiples of

.
2
Suppose the terminal side of angle θ in standard position intersects the unit circle at the
point (x, y). Then the ratio
y
is the tangent of θ, denoted tan θ.
x
Properties of Tangent Functions:Suppose y = a tan bθ,with a ≠ 0, b > 0 and θ in radians:

b
is the period of the function;
One cycle occurs in the interval from 

2b
to
There are vertical asymptotes at each end of a cycle.
You can use asymptotes and 3 points to sketch a tangent curve. The 5 elements are
equally spaced through one cycle. Use the pattern is asym-(-a)-zero-(a)-asym.

2b
;
Algebra 2 Section 13.7 Notes
Translating Sine & Cosine Functions
You can translate periodic functions in the same way that you translate other functions.
Each horizontal translation of certain periodic functions is a phase shift.
Families of Sine & Cosine Functions:
Parent Function
Transformed Function
y = sin x
y = a sin b(x – h) + k
y = cos x
y = a cos b(x – h) + k
|a| = amplitude (vertical stretch or shrink)
2
= period (when x is in radians and b > 0)
b
h = phase shift, or horizontal shift
k = vertical shift (y = k is the midline)
Algebra 2 Section 13.8 Notes
Reciprocal Trigonometric Functions
To solve an equation ax = b, you multiply each side by the reciprocal of a. If a is a
trigonometric expression, you need to use its reciprocal.
Cosine, sine, and tangent have reciprocals. Cosine and secant are reciprocals, as are sine
and cosecant. Tangent and cotangent are also reciprocals.
The cosecant (csc), secant (sec), and cotangent (cot) functions are defined using
reciprocals. Their domains do not include the real numbers θ that make the
denominator zero.
csc  
1
sin 
(cot θ = 0 at odd multiples of
sec  
1
cos 
cot  

, where tan θ is undefined.)
2
1
tan 
Algebra 2 Section 13.7 Notes
Translating Sine & Cosine Functions
You can translate periodic functions in the same way that you translate other functions.
Each horizontal translation of certain periodic functions is a phase shift.
Families of Sine & Cosine Functions:
Parent Function
Transformed Function
y = sin x
y = a sin b(x – h) + k
y = cos x
y = a cos b(x – h) + k
|a| = amplitude (vertical stretch or shrink)
2
= period (when x is in radians and b > 0)
b
h = phase shift, or horizontal shift
k = vertical shift (y = k is the midline)
Algebra 2 Section 13.8 Notes
Reciprocal Trigonometric Functions
To solve an equation ax = b, you multiply each side by the reciprocal of a. If a is a
trigonometric expression, you need to use its reciprocal.
Cosine, sine, and tangent have reciprocals. Cosine and secant are reciprocals, as are sine
and cosecant. Tangent and cotangent are also reciprocals.
The cosecant (csc), secant (sec), and cotangent (cot) functions are defined using
reciprocals. Their domains do not include the real numbers θ that make the
denominator zero.
csc  
1
sin 
(cot θ = 0 at odd multiples of
sec  
1
cos 
cot  

, where tan θ is undefined.)
2
1
tan 