4/16/13 Section 4.3

Obj: SWBAT apply properties of

angles and their measures.

Bell Ringer:

ACT Problem See overhead

Parking lot #37, 39, 41, 72

HW Requests: pg 369 #41-47 odds pg 383 #1, 2, 3-11 odds

Homework: From the Textbook WS pg 383 #13-47 odds.

Announcements:

Report Card Pick up on Thursday

Education is Power!

Special Right Triangles (Reference)

n

n

2n

45

45 1

1

2

45

45

This is our reference triangle for the 45-45-90.

n 2n

30

60

3n

1 2

30

60

3This is our reference triangle for the 30-60-90.

We can find coterminal angles. Not let us move on to the coordinate

plane. Let Ө be an acute angle in standard position whose terminal side

contains the point, (2,3). Find the six trig functions.

Ө

5

Definitions of Trigonometric Functions of Any Angle

Let be an angle in standard position with (x, y) a point on the

terminal side of and

Definitions of Trig Functions of Any Angle

(Sect 8.1)

2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r x

y x

x y

y

x

(x, y)

r

6

Since the radius is always positive (r > 0), the signs of the

trig functions are dependent upon the signs of x and y.

Therefore, we can determine the sign of the functions by

knowing the quadrant in which the terminal side of the

angle lies.

The Signs of the Trig Functions

7

The Signs of the Trig Functions

8

Where each trig function is POSITIVE:

A

C T

S

“All Students Take Calculus”

Translation:

A = All 3 functions are positive in Quad 1

S= Sine function is positive in Quad 2

T= Tangent function is positive in Quad 3

C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is positive,

but sine and cosine are negative; in Quad 4, cosine is positive but sine and tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive,

secant is positive wherever cosine is positive, and cotangent is positive wherever tangent is

positive.

9

Where each trig function is POSITIVE:

A

C T

S

“All Students Take Calculus”

Translation:

A = All 3 functions are positive in Quad 1

S= Sine function is positive in Quad 2

T= Tangent function is positive in Quad 3

C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3,

tangent is positive, but sine and cosine are negative; in Quad 4, cosine is

positive but sine and tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever

sine is positive, secant is positive wherever cosine is positive, and cotangent is

positive wherever tangent is positive.

II I

III IV

11

Determine if the following functions are positive or negative:

Example

sin 210°

cos 320°

cot (-135°)

csc 500°

tan 315°

12

Examples

• For the given values, determine the quadrant(s) in which

the terminal side of θ lies.

1) sin 0.3614 2) tan 2.553 3) cos 0.866

13

Examples

• Determine the quadrant in which the terminal side of θ lies,

subject to both given conditions.

1) sin 0, cos 0 2) sec 0, cot 0

14

Examples

• Find the exact value of the six trigonometric functions of θ

if the terminal side of θ passes through point (3, -5).

15

The values of the trig functions for non-acute angles (Quads II, III, IV) can

be found using the values of the corresponding reference angles.

Reference Angles (Sect 8.2)

Definition of Reference Angle

Let be an angle in standard position. Its reference angle

is the acute angle formed by the terminal side of and

the horizontal axis.

ref

16

Example

Find the reference angle for 225

Solution y

x

ref

By sketching in standard position, we

see that it is a 3rd quadrant angle. To find

, you would subtract 180° from 225 °. ref

225 180

45

ref

ref

17

So what’s so great about reference angles?

Well…to find the value of the trig function of any non-acute angle, we

just need to find the trig function of the reference angle and then

determine whether it is positive or negative, depending upon the

quadrant in which the angle lies.

For example,

1sin 225 (sin 45 )

2

45° is the ref

angle

In Quad 3, sin is

negative

Find the reference angle

• Important idea: the triangle is always drawn to the x axis

• The reference angle is the POSITIVE ACUTE angle made in the triangle closest to the origin

x

y

’

’

Find the reference angle

• Find the reference angle for 288

• Draw 288 angle

• Draw triangle back to x axis

• Find reference angle x

y

’

• ’ is what is left over from 360

• 360 – 288 = ’

• = 72

Find the reference angle

• Find the reference angle for 98

• Draw 98 angle

• Draw triangle back to x axis

• Find reference angle x

y

’

• ’ is what is left over from 180

• 180 – 98 = ’

• = 82

Find the reference angle

• Find the reference angle for -55

• Draw -55 angle

• Draw triangle back to x axis

• Find reference angle x

y

’

• ’ is what is the same as measured from zero but the positive angle

• |-55| = ’

• = 55

22

In general, for in

radians,

A second way to measure angles is in radians.

Radian Measure (Sect 8.3)

s

r

Definition of Radian:

One radian is the measure of a central angle that intercepts

arc s equal in length to the radius r of the circle.

23

Radian Measure

2 radians corresponds to 360

radians corresponds to 180

radians corresponds to 902

2 6.28

3.14

1.572

24

Radian Measure

25

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

180

180

Example

Convert from degrees to radians: 210º

210

Angle- formed by rotating a ray

about its endpoint (vertex)

Initial Side Starting position

Terminal Side Ending position

Standard Position Initial side on positive x-axis

and the vertex is on the origin

An angle describes the amount and direction of rotation

120° –210°

Positive Angle- rotates counter-clockwise (CCW)

Negative Angle- rotates clockwise (CW)

Coterminal Angles: Two angles with the same initial

and terminal sides

Find a positive coterminal angle to 20º 38036020

34036020Find a negative coterminal angle to 20º

Types of questions you will be asked:

Identify a) ALL angles coterminal with 45º, then b) find one

positive coterminal angle and one negative coterminal angle.

a) 45º + 360k (where k is any given integer).

b) Some possible answers are 405º, 765º, - 315º, - 675º

To find a coterminal angle add or subtract multiples of 360º

if in degrees or 2π if in radians

Coterminal Angles: Two angles with the same initial

and terminal sides

Find a positive coterminal angle to 60º 42036060

30036060

Find 2 coterminal angles to 4

15

4

8

4

15

2

4

15

4

8

4

15

2

4

15

4

23

4

8

4

7

Find a negative coterminal angle to 60º

4

Degree - angular measure equal to 1/180th of a straight angle

Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

SOH-CAH-TOA