Section 2.1 Angles and Their Measure. Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle.

Section 2.1 Angles and Their Measure

Vertex Initial Side

Terminal si

de

Counterclockwise rotationPositive Angle

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

Relationship between Degrees and Radians

-> 1 revolution = 2 π radians

-> 180o = π radians

1801

1801

radian

radian

Announcements• Test Friday (Jan 30) in lab, ARM

213/219, material through section 2.2• Sample test posted...link from course

Web site• Bring picture ID…you will need to

scan your ID upon entering the lab• You may use a calculator up to TI 86.• You can’t your the book or notes

Section 2.2Right Angle Trigonometry

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

/

/

/

/

/

/

Pathagorean Theorem

a

bc a2 + b2 = c2

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

Pythagorean Identities

The equation

sin2θ + cos2 θ

along with

tan2 θ + 1 = sec2 θ

and

1 + cot2 θ = csc2 θ

are called the Pythagorean identities.

More Identities

tan

1cot

cos

1sec

sin

1csc

sin

coscot

cos

sintan

Reciprocal Identities

Quotient Identities

Complementary Angles Theorem

Cofunctions of complementary angles are equal.

Two acute angles are complementary if the sum of their measures is a right angle…90 degrees.

α

β

The angles α and β are complementary in a right triangle, α + β = 90 degrees.

Complementary Angles in Right Triangles

Cofunctions

)90sec(csc)90csc(sec

)90tan(cot)90cot(tan

)90sin(cos)90cos(sin

Degrees

Radians

2seccsc

2cscsec

2tancot

2cottan

2sincos

2cossin

Using the Complementary Angle Theorem

Find the exact value (no calculator) of the following expressions.

70cos20cos1.

50sin

40cos.

22 b

a

50sin

40cos.a

50sin)4090sin(40cos

150sin

50sin

50sin

40cos

70cos20cos1. 22 b

70sin20cos 22

70cos70sin1 22

)70cos70(sin1 22

0)1(1

so