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TRIGONOMETRY
5.1: ANGLES AND THEIR MEASURES:
Terms:
1. Two rays with a common vertex form an angle.
2. One of the rays of the angle is called the initial side and the other ray is
called the terminal side.
3. Angles are identified by showing direction and amount of rotation.
4. Positive angles are depicted with a counterclockwise direction.
5. Negative angles are depicted with a clockwise rotation.
Examples:
6. Angles are usually denoted by the Greek letters alpha ( ), beta ( ), gamma
( ), and theta ( ).
7. An angle is said to be in standard position if its vertex is at the origin of a
rectangular coordinate system and its initial side coincides with the positive half
of the x-axis.
8. An angle in standard position whose terminal side lies in a quadrant is said to
be an angle in that quadrant.
9. An angle in standard position whose terminal side lies on the x- or y-axis is
called a quadrantal.
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10. Angles are measured by determining the amount of rotation needed for the
initial side to become coincident with the terminal side.
11. Two common unit of measurement: degree and radian
12. a. 1 counterclockwise direction is the equivalent of 360 degrees.
b. One degree is therefore 1 / 360 of a revolution.
c. An angle that measures ¼ of a revolution is called a right
angle.
d. An angle that measures ½ of a revolution is 180 degrees.
e. An angle that measures ¾ of a revolution is 270 degrees.
RECALL:
9 A central angle is an angle whose vertex is at the center of the
circle.
9 The rays of the central angle intersect an arc of the circle.
13. If the radius of the circle is r and the length of the arc intersected
by the central angle is also r, then the measure of the angle is
1 radian.
RELATIONSHIP BETWEEN DEGREES AND RADIANS:
9 1 revolution = 2 radians = 360 degrees
9 ½ revolution =
radians = 180 degrees
9 ¼ revolution =
radians = 90 degrees
CONVERSIONS:
9 To convert from degrees to radians multiply by / 180
9 To convert from radians to degrees multiply by 180 /
MEMORIZE: (PAGE 375)
Table 1 lists the degree and radian measures of some commonly encountered
angles. Learn both the degree and radian measure and in which quadrant or axis
the angle lies.
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5.2: TRIGONOMETRIC FUNCTIONS
UNIT
CIRCLE
RECALL: The unit circle is a circle whose radius is 1 and whose center is at the
origin of a rectangular coordinate system.
Note: For any real number t, we can locate a unique point P(x,y) on the unit
circle. The coordinates of the point P(x,y) on the unit circle are used to define the
six trigonometric functions.
SIX TRIGONOMETRIC FUNCTIONS: Let t be a real number and let
P(x,y) be the point on the unit circle that corresponds to t:
9 The sine function is defined as sin t = y
9 The cosine function is defined as cos t = x
9 The tangent function is defined as tan t = y / x
9 The cosecant function is defined as csc t = 1 / y
9 The secant function is defined as sec t = 1 / x
9 The cotangent function is defined as cot t = x / y
Notes:
1. The sine and cosecant functions are reciprocal functions.
2. The cosine and secant functions are reciprocal functions.
3. The tangent and cotangent functions are reciprocal functions.
4. The six trigonometric functions are defined for angles in
standard position.
THINGS YOU NEED TO KNOW:
FIND THE EXACT VALUES FOR QUADRANTAL ANGLES (table 2) AND
FOR SPECIAL ANGLES (45, 30 AND 60) (table 3)
USE A CALCULATOR TO EVALUATE TRIGONOMETRIC FUNCTIONS
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EVALUATE THE TRIGONOMETRIC FUNCTIONS FOR CIRCLES WHOSE
RADIUS IS r.
HAPPY VALENTINE’S
DAY
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