Download Lecture 5 (L5): Trig Functions of Any Angle and Unit Circle Textbook

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Lecture 5 (L5): Trig Functions of Any Angle and Unit Circle
Textbook Section: 4.4 and 4.2
Review:
Six trigonometric functions for an acute angle, 𝜽:
Problem: Consider the obtuse angle,
5𝜋
6
, in standard position on the coordinate plane. Let
(−2√3, 2) be a point on the terminal side of the angle. How can we define the six
trigonometric functions for the angle
5𝜋
6
?
Reference Angles:
If 𝜃 is an angle in standard position on the coordinate plane, then its
_____________________________________________ is the acute angle formed by 𝜃′𝑠 terminal side and
the ________________________________________. These angles can be used to find the values of the
six trigonometric functions for an angle 𝜃 in standard position.
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Problem: Find the sine, cosine, and tangent values for the angles
5𝜋
4
and
5𝜋
3
.
Problem: Consider an angle, 𝜃, in standard position that does not have a special angle as a
reference angle. How can the six trigonometric functions be defined for any angle? Let 𝜃 be
any angle in standard position and (𝑥 , 𝑦) be a point on its terminal side.
Problem: How does the quadrant affect the sign of the trigonometric function values?
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Review:
Equation of a Circle with center (h, k) and radius, r:
Unit Circle:
The ______________________________ is a circle centered at the _______________________ with a
_____________________ of one.
Diagram and Algebraic Representation:
Recall how radians were defined as the intercepted angle of a circle with radius, r,
and arc length, s, as ___________________________________.
Imagine the real number line wrapping around the unit circle. Each number
on the number line would correspond with a _______________________________ on the plane. Each
number on the number line also corresponds to a central angle of the unit circle. Recall the
formula for arc length, s, is _____________________________________. So each point along the circle
is an ___________________________________ intercepted by a ______________________________ in radians.
Diagram:
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Trigonometric Functions of Real Numbers:
Since each real number along the unit circle is defined as ____________________________
with radius being ______________ , we can define the trigonometric functions for each real
number along the unit circle. Remember each real number along the unit circle also
represents a ______________________________ on the coordinate plane.
Definitions for Trigonometric Functions:
Let t represent a real number along the unit circle and (𝑥, 𝑦) is the point
corresponding to the real number, t, along the unit circle. Then the trigonometric functions
for the real number, t, are defined as follows:
CAUTION!! TRIG FUNCTIONS ARE FUNCTIONS OF ANGLES!! ON THE UNIT
CIRCLE BECAUSE THE RADIUS IS ONE THE ARC LENGTH EQUALS THE
ANGLE MEASURE. THIS WILL NOT ALWAYS HAPPEN!!!!!!!!!!!!
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Problem: Consider the real number 𝑡 =
𝜋
6
on the unit circle. What coordinate, (𝑥, 𝑦), does
t correspond to?
Problem: What are the corresponding coordinates along the unit circle for the real
numbers: 0 ,
𝜋
2
,𝜋 ,
3𝜋
2
, 2𝜋 .
Problem: Determine the exact values of the trigonometric functions of the real number t,
3
4
5
5
given that t corresponds to the point ( , − ).
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The unit circle can be divided into arcs corresponding to the real number t values of:
Diagram:
Domain and Range of Sine and Cosine Functions:
We discussed earlier that sine and cosine could never be greater than
__________________. Then a lower bound for sine and cosine is ___________________________.
This means that the _____________________________ of sine and cosine is
_____________________________________.
Now we consider that the unit circle is made by wrapping the real line around the
circle. The sine and cosine functions can continue take values as we continue to move
around the circle. That means that the _____________________________ of the sine and cosine
functions is ________________________________________.
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Period of the Sine and Cosine Functions:
A function is said to be _________________________________ if there is a positive real
number, a, such that ______________________________________________ for all x in the domain of the
function. The smallest number, a, such that the function is periodic is called the
_______________________________.
Problem: What is the period of the sine and cosine functions?
Review:
Odd function:
Even function:
Problem: Are the six trigonometric functions odd or even?
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