MATH 2312 - Section 4.3 - Trigonometric Functions of ...
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MATH 2312 - Section 4.3 - Trigonometric Functions of Angles
An angle is in standard position if the vertex is at the origin of the two-dimensional plane
and its initial side lies along the positive x-axis. Positive angles are generated by
counterclockwise rotation. Negative angles are generated by clockwise rotation.
An angle in standard position whose terminal side lies on either the x-axis or the y-axis is
called a quadrantal angle.
The Reference Angle or Reference Number
Let ? be an angle in standard position. The reference angle associated with ? is the acute
angle (with positive measure) formed by the ?-axis and the terminal side of the angle ?.
When radian measure is used, the reference angle is sometimes referred to as the reference
number (because a radian angle measure is a real number).
? 2021 I. Perepelitsa
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Example: Draw each angle in standard position and specify the reference angle.
a.
?120¡ã
Reference angle:
b.
7?
4
Reference angle:
Angles that terminate in the exact same position are called coterminal angles.
Every angle has infinitely many coterminal angles.
An angle of ? ¡ã is coterminal with angles ? ¡ã + 360¡ã ?, where k is an integer.
An angle of ? radians is coterminal with angles ? + 2??, where k is an integer.
Example: Find a positive and negative angle that is coterminal with
5?
6
.
? 2021 I. Perepelitsa
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We previously defined the six trigonometric functions of an angle as ratios of the length of
the sides of a right triangle. Now we will look at them using a unit circle centered at the
origin in the coordinate plane. This circle will have the equation ? 2 + ? 2 = 1.
If we select a point ?(?, ?) on the circle and draw a ray from the origin though the point, we
have created an angle in standard position.
Trigonometric Functions of Angles
Let us zoom in:
P(x, y)
The circle above is the unit circle so ? = 1. Use the triangle and SOH-CAH-TOA, to obtain
the following:
cos ? = ?
1
sec ? = ? , ? ¡Ù 0
sin ? = ?
1
csc ? = ? , ? ¡Ù 0
?
tan ? = ? , ? ¡Ù 0
?
cot ? = ? , ? ¡Ù 0
So for the point ?(?, ?) = (cos ? , sin ?).
Example: Let the point ?(?, ?) denote the point where the terminal side of angle ? (in
standard position) meets the unit circle. ? is in Quadrant III and ? = ? 5?13. Evaluate the
six trig functions of ?.
? 2021 I. Perepelitsa
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3?
Example: For the quadrantal angle , give the coordinates of the point where the
2
terminal side of the angle interests the unit circle. Then find cosine, tangent and cosecant,
if possible, of the angle.
Recall: (x, y) = (cos ? , sin ? )
y
(-, +)
(-, -)
(+, +)
All Students Take Calculus
y
x
OR
(+, -)
S
A
T
C
x
This should help you to know which trigonometric functions are positive in which
quadrant.
Example: Name the quadrant in which the given conditions are satisfied.
a. tan ? > 0, sin ? < 0
Quadrant
b. sec ? > 0, cot ? > 0
Quadrant
? 2021 I. Perepelitsa
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Example: Rewrite each expression in terms of its reference angle, deciding on the
appropriate sign.
7?
a. sin(140¡ã )
b. cos ( 6 )
c. sec (?
5?
3
)
? 2021 I. Perepelitsa
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