An angle is formed by two rays that have a common endpoint. One ray is called the

initial side and the other the terminal side.

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system 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 is called a quadrantal angle if its terminal side lies on the x-axis or on the y-axis.

Angle is an example of a quadrantal angle.

Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360°.

An acute angle measures less than 90°.

A right angle measures 90°.

An obtuse angle measures more than 90° but less than 180°.

A straight angle measures 180°.

90

360

270

180 0

Complementary Angles:

Supplementary Angles:

Find the complement and supplement angles of 40o.

The sum of any two angles that equals 90o.

The sum of any two angles that equals 180o.

90o – 40o = 50o 180o – 40o = 140o

901216 xx

means 90o

901216 xx

9028 x

888 x

11x

180259117 yy1803616 y14416 y

9y

There are 60 minutes in 1 degree. '601

60

1'1

There are 60 seconds in 1 minute. "60'1

3600

1

60

1"1

'

'4632

'7583

75 min. > 60 min., so carry 1 degree.

'1584'601

We need to borrow 1 degree and convert it to 60 min.

'3718'6089 '3718'2371

3600

14

60

821

Convert the minutes and seconds to fractional degrees.

Calculator help!

Round your answer according to the directions.

The whole number is the degrees.Multiply the decimal by 60 to determine the minutes.

'602678.034

'068.1634The whole number is the minutes.Multiply the decimal by 60 to determine the seconds.

"4'1634

21.13722222

Two angles with the same initial and terminal sides butpossibly different rotations are called coterminal angles.

Increasing or decreasing the angle measure of an angle in standard position by an integer multiple of 360o results in a coterminal angle. Thus, an angle of is coterminal with angles of , where k is an integer.

k 360

Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following:

a. a 400° angle b. a –855° angle 40360400 2253360855

CD players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through how many degrees will a point on the edge of the CD move in 2 seconds?

Determine how many revolutions in 1 second. 860

480

8 revolutions in 1 second times 2 is 16 revolutions.

16 revolutions times 360o. 576036016

In the diagram, to the right, 2 parallel lines are intersected by a transversal line.

Are formed by 2 intersecting lines and are the angles that open opposite of each other.Name all the pairs of vertical angles.

1 & 4, 2 & 3, 5 & 8, and 6 & 7.

Are the matching angles formed by the two intersections of the parallel lines.Name all the pairs of corresponding angles. 1 & 5, 2 & 6, 3 & 7, and 4 & 8.

Vertical angles are always equal in measure.

Corresponding angles are always equal in measure.

Are the angles in between the parallel lines and alternate over the transversal line.

Symbol for parallel.

Name all the pairs of alternate interior angles. 3 & 6 and 4 & 5.Alternate interior angles are always equal in measure.

Are the angles outside the parallel lines and alternate over the transversal line.Name all the pairs of alternate exterior angles. 1 & 8 and 2 & 7.

Alternate exterior angles are always equal in measure.

Are the angles in between the parallel lines and on the same side of the transversal line.Name all the pairs of same side interior angles. 3 & 5 and 4 & 6.

Same side interior angles are always supplementary.

Sum of the angles of a triangle_______________________________.always adds up to 180 degrees

1 2

3

All three angles are acute, less than 90o.

One angle is 90o.

One angle is greater than 90o.

All three sides are equal.

Two sides are equal.

No sides are equal.

Two triangles that are the same shape, but not necessarily the same size.

1. Corresponding angles must be the same measure.

2. Corresponding sides must be proportional.

FC

EB

DA

DEFABC ~

FE

DCB

A

FD

CA

EF

BC

DE

AB1. 2.

Examples.Find the measure of all angles. Find the measure of all angles and sides.

12

34 5 75

76 9140

810 11 12

A

B C

R

S T

1064712

3

16

6

75

140

40

40

4040

654075180

65

65

7575 105

105

ABC ~ RST

RA SB TC 47A

47

S106

106

TC 10647180

TC 27

2727

TR

CA

ST

BC

RS

AB1612

3 6

ST

16

1

444116 ST

4

61

4 CA

24164 CA

24

Definitions of Trigonometric Functions of Any Angle.

Let be any angle in standard position and let P = (x, y) be a point on the terminal side of . If is the distance from (0, 0) to (x, y), the six trigonometric functions of are defined by the following ratios:

2 2r x y

sinyr

cosxr

tan , 0yx

x

csc , 0ry

y

sec , 0rx

x

cot , 0xy

y

What do you notice about the fractions? Reciprocals!

Pythagorean Theorem222 yxr

Let P = (8, 15) be a point on the terminal side of . Find each of the six trigonometric functions of . P = (8, 15) is a point on the terminal side of . x = 8 and y = 15.

r

ysin

r

xcos

x

ytan

2 2r x y

17

15

17

8

8

15

y

xcot

15

8

x

rsec

8

17

y

rcsc

15

17

15,8P

8x

15y

22 158 22564 17289

17r

Let P = (1, –3) be a point on the terminal side of . Find each of the six trigonometric functions of . P = (1, –3) is a point on the terminal side of . x = 1 and y = –3.

r

ysin

r

xcos

x

ytan

3,1 P

1x3y

2 2r x y 2 2(1) ( 3) 1 9 10

10

3

10

10

10

103

10

1

10

10

10

10

31

3

y

xcot

3

1

3

1

x

rsec 10

1

10

y

rcsc

3

10

3

10

10r

Find the six trigonometric function values of the angle in standard position, if the terminal side of is defined by 3x + 4y = 0, x < 0.

x < 0 means that the terminal side is in quadrant 2 or 3!

043 yx

Solve the linear equation for so it is in slope intercept form, y = mx + b.

034 xy

04

3

xy

Start at (0, 0) and slope of -3/4.

Since we are in quadrant 2 or 3, we will have to reverse the slope…up 3, left 4.

3,4

3,4 yx

525916

34 22

r

5

3sin

r

y

5

4cos

r

x

4

3tan

x

y 3

4cot

y

x

4

5sec

x

r

3

5csc

y

r

Section 1.4 Reciprocal Identities.

sin

r

xcos

x

ytan

y

xcot

x

rsec

y

r

1

csc

Flip the fractions

r

y

csc

1

csc

1

y

rcsc

Use this same concept for the other trig. functions we get the rest of the identities

r

y

sin

1csc

sec

1

cos

1

cot

1 tan

1

ryyr

We know that r is always positive andx < r & y < r.

sin & cscSin & csc

sin & csc sin & cscrx

xr

cos & sec

Cos & sec

cos & sec

cos & sec

xyyx

tan & cot

Tan & cot

tan & cot

tan & cot

AAll

SStudents

TTake

CCalculus

siny cosy

tany coty

secy cscy

ry

rx

Since x and y are < r, the fraction < 1. If x and y are negative, the fraction > -1.

11 y 1,1

xy

yx

Since x and y can be all real numbers, dividing real numbers will still be real numbers. ,

xr

yr

Since x and y are < r, the fraction > 1. If x and y are negative, the fraction < -1.

1y ,11,

Use All Students Take Calculus

Negative

AS

T C

++–

–

Quadrant 2 has the sign representation for both conditions, therefore, the terminal side is in quadrant 2.

Positive

23

3

2sin

r

y

222 ryx Use the Pythagorean Theorem.

222 32 x

942 x

52 x

5xx has to be negative because we are in quadrant 2.

r

xcos

x

ytan

y

xcot

x

rsec

y

rcsc

3

5

5

2

5

52

2

5

5

3

5

53

2

3

5

Divide by r2 Divide by y2Divide by x2

12

2

2

2

r

y

r

x

2r2r2r

122

r

y

r

x

1sincos 22

1sincos 22

2x2x2x 2y2y2y

2

2

2

2

1x

r

x

y

22

1

x

r

x

y

22 sectan1

22 sectan1

2

2

2

2

1y

r

y

x

22

1

y

r

y

x

22 csc1cot

22 csc1cot

rx

ry

x

y tan cot

AS

T C

++–

–

1cossin 22 4

3

Negative Positive

4

3cos

r

x

22243 y

13

1632

2

y

y

Since we flipped the sine and cosine, the reciprocal of tangent is cotangent.

13yPositive, Quad. 2

r

ysin

4

13

x

ytan

3

13

3

3

3

13

3

39

Using identities may be quicker.Pythagorean and Quotient.

1sin2

432

1sin 1632

16132sin

413sin

4

3

413

cos

sintan

3

39

AS

T C4

Negative

3

Positive

x

ytan

3

4

Negative

– 3

– 4 r

22 yxr

22 43 r

525169 r

r

ysin

5

4

r

xcos

5

3

22 sec1tan Using Pythagorean & Reciprocal Identities.

22

34 sec1

29

16 sec1

2925 sec

sec35

cos53

22 cos1sin

2

531sin 5

4

54sin