Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle.
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Transcript of Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle.
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Vertex Initial Side
Terminal si
de
Counterclockwise rotationPositive Angle
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Vertex Initial Side
Terminal si
de
Clockwise rotationNegative Angle
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An angle is said to be in
if its vertex is at the origin of a rectangular
coordinate system and its initial side
coincides with with positive - axis.
standard position
x
Initial sideVertex
Terminal side
x
y
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The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated 360.
Initial sideTerminal side
Vertex
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Initial side
Terminal side
Vertex
90 angle; 14
revolution
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Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.
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r
r
1 radian
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Theorem Arc Length
For a circle of radius r, a central angle of radians subtends an arc whose length s is
s r
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A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle.
cb
a
90
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Initial side
Terminal side
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a
bc
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The six ratios of a right triangle are called trigonometric functions of acute angles and are defined as follows:
sine of
cosine of
tangent of
cosecant of
secant of
cotangent of
Function namesin
cos
tan
csc
sec
cot
Abbreviation Valueb c
a c
b a
c b
c a
a b
/
/
/
/
/
/
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Find the value of each of the six trigonometric functions of the angle .
Adjacent
12 13
c = Hypotenuse = 13
b = Opposite = 12a b c2 2 2
a2 2 212 13
a2 169 144 25 a 5
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a
b
c
Adjacent = 5
Opposite =
Hypotenuse =
12
13
sin OppositeHypotenuse
1213
cos AdjacentHypotenuse
513
tan OppositeAdjacent
125
csc HypotenuseOpposite
1312
sec HypotenuseAdacent
135
cot AdjacentOpposite
512
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b a c2 2 2
cb
a
90
b
c
a
c
c
c
2
2
2
2
2
2
bc
ac
2 2
1
sin cos2 2 1
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Theorem Complementary Angles Theorem
Cofunctions of complementary angles are equal.