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February 02, 2017
Section 4.3
Name________________________________
Trigonometry Extended: The Circular Functions
An angle is in standard position if its initial side lies on the
positive x-axis and its terminal side is found by rotating a copy of
this side either clockwise or counterclockwise with the vertex of
the angle at the origin. A counter-clockwise rotation results in an
angle with positive measure and a clockwise rotation results in an
angle with negative measure. Two angles with the same initial
and terminal sides are called coterminal angles. The measures of
coterminal angles differ by an integer multiple of 2π or 360°.
rm
te
al
n
i
e
is d
2
initial side
1
,
0≤
1 , and
2
are ___________ angles.
< 2π; 2π ≤
< 4π; -2π ≤
< 0.
We will primarily use radian measure when
working with the circular functions.
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The Unit Circle is a circle of radius 1 centered at the origin.
What is the equation of the unit circle? _______________
The Unit Circle Definitions of the Trig Functions
Let t be any real number and let P(t) = (x, y) be the
point on the unit circle corresponding to t. Then
for any angle t,
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P(t) = (x, y)
t
The Trigonometric Functions of any Angle
Let θ be any angle in standard position and let P(x,y) be
any point on the terminal side of the angle. Let r denote
the distance from P(x,y) to the origin.
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The point (-4,3) is on the terminal side of angle θ. Evaluate
the six trigonometric functions for θ.
(-4,3)
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The point (5, -2) is on the terminal side of the angle θ.
Evaluate the six trigonometric functions for θ.
We can draw a figure like the one for the problem about if we wish, but it is unnecessary.
We only need to calculate the value of r using the given x and y coordinates and then use
the circular definitions for the trig functions of θ.
In what quadrant does the terminal side of θ lie? ______
Which two of the six trig functions have positive values
for this quadrant? ______________________________
Do our answers agree with this? _____
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February 02, 2017
Lesson 4.3 Homework 1 - Page 347 #6, 8, 13-20.
#6 Evaluate the six trigonometric functions of the angle θ.
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is on the terminal side of the angle θ.
#8 The point
Evaluate the six trig functions for θ.
State the sign of each trig function in the interval given.
sin t and csc t cos t and sec t tan t and cot t
Determine the sign of each trig function and state the quadrant
in which the terminal side of the angle lies.
cos 143°
tan 192°
Sign
Quadrant
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The 16-Point Unit Circle
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What is the measure in radians of an angle that is onefourth of a full rotation?
Degrees?
What is the measure in radians of an angle that is oneeighth of a full rotation?
Degrees?
What is the measure in radians of an angle that is onetwelfth of a full rotation?
Degrees?
What are the measures of the Quadrantal Angles in
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radians?
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Lesson 4.3 Homework 2
Evaluate without using a calculator by using the 16-Point Unit Circle.
Choose the point on the terminal side of θ for each.
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Reference Triangles
Quadrant I
Quadrant III
Quadrant IV
In which quadrant
should we drawn the reference triangle?
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Lesson 4.3 Homework 3 - Page 348 #45, 48, 49 & 52.
#45
#48
The sine and cosine functions are periodic functions with a period
of 2π since
and
for all
integers n. The period of the tangent function is π, therefore
for all integers n.
#49
#52
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