Vertex Initial Side

Terminal si

de

Counterclockwise rotationPositive Angle

Vertex Initial Side

Terminal si

de

Clockwise rotationNegative Angle

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

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

Initial side

Terminal side

Vertex

90 angle; 14

revolution

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.

r

r

1 radian

Theorem Arc Length

For a circle of radius r, a central angle of radians subtends an arc whose length s is

s r

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

Initial side

Terminal side

a

bc

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

/

/

/

/

/

/

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

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

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

Theorem Complementary Angles Theorem

Cofunctions of complementary angles are equal.