warm-up 3/ find the measure of

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Warm-Up 3/27-28 1. Find the measure of . sin = 14 16 =sin โˆ’ 1 ( 14 16 ) =61 ยฐ

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Page 1: Warm-Up 3/ Find the measure of

Warm-Up 3/27-281. Find the measure of .

sin ๐œƒ=1416

๐œƒ=sinโˆ’1( 1416 )

๐œƒ=61ยฐ

Page 2: Warm-Up 3/ Find the measure of
Page 3: Warm-Up 3/ Find the measure of

Rigor:You will learn how to find reference

angles, to evaluate and determine sign of trig functions of any angle.

Relevance:You will be able to solve real world problems using reference angles.

Page 4: Warm-Up 3/ Find the measure of

Trig 3: Trigonometric

Functions on the Unit Circle

Page 5: Warm-Up 3/ Find the measure of

ฮธ

(x, y)

r y

x

sin ๐œƒ=ยฟ ๐‘ฆ๐‘Ÿ

cos๐œƒ=ยฟ๐‘ฅ๐‘Ÿ

tan๐œƒ=ยฟ ๐‘ฆ๐‘ฅ, ๐‘ฅโ‰ 0

๐‘Ÿ=โˆš๐‘ฅ2+๐‘ฆ2

Page 6: Warm-Up 3/ Find the measure of

Trigonometric Functions of Any Angle

Page 7: Warm-Up 3/ Find the measure of

Example 1: Let (8, โ€“6) be a point on the terminal side of and angle in standard position. Find the exact values of the three basic trig functions of .

s ๐‘–๐‘›๐œƒ=๐‘ฆ๐‘Ÿ ยฟโˆ’610

cos๐œƒ=๐‘ฅ๐‘Ÿ ยฟ

810

ta๐‘›๐œƒ=๐‘ฆ๐‘ฅ ยฟโˆ’68

ยฟโˆ’ 35

ยฟ45

ยฟโˆ’ 34

๐‘Ÿ=โˆš82+ (โˆ’6 )2=โˆš100๐‘Ÿ=10

8

โ€“ 6 10

โ€™

Page 8: Warm-Up 3/ Find the measure of

โ€ข For 0ยฐ or 360ยฐ, use the coordinate (1, 0)โ€ข For 90ยฐ, use the coordinate (0, 1)โ€ข For 180ยฐ, use the coordinate (โ€“ 1, 0)โ€ข For 270ยฐ, use the coordinate (0, โ€“ 1 )

2

32

0 or 2r = 1

Page 9: Warm-Up 3/ Find the measure of

Example 2: Find the exact value of each trigonometric function, if defined. If not defined, write undefined.

a. sin(โ€“ 180)

b. tan

P(โ€“ 1 , 0), r = 1

ยฟ๐‘ฆ๐‘Ÿยฟ

01ยฟ0

P(0 , โ€“ 1), r = 1

ยฟ๐‘ฆ๐‘ฅ ยฟโˆ’10 ๐‘ข๐‘›๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘’๐‘‘

Page 10: Warm-Up 3/ Find the measure of

Reference Angle: an acute angle formed by the terminal side and the x-axis.

Page 11: Warm-Up 3/ Find the measure of

Example 3: Sketch each angle. Then find the reference angle.

a. โ€“ 150

b.

x

y

x

y

= โ€“ 150 180 โ€“ 150 = 30 โ€™= 30

๐œƒ=3๐œ‹4โ€™=

4๐œ‹4 โˆ’ 3๐œ‹

4 =๐œ‹4

Page 12: Warm-Up 3/ Find the measure of

ฮธ

(x, y)

r y

x

sin ๐œƒ=ยฟ๐‘ฆ๐‘Ÿ

cos๐œƒ=ยฟ๐‘ฅ๐‘Ÿ

tan๐œƒ=ยฟ๐‘ฆ๐‘ฅ, ๐‘ฅโ‰ 0

๐‘Ÿ=1sin ๐œƒ=๐‘ฆcos๐œƒ=๐‘ฅ

(cos, sin)๐‘Ÿ=โˆš๐‘ฅ2+ ๐‘ฆ2

tan๐œƒ=sin ๐œƒcos๐œƒ

Page 13: Warm-Up 3/ Find the measure of

x

y

Quadrant IQuadrant II

Quadrant III Quadrant IV

(+x, +y)(โ€“x , +y)

(โ€“x , โ€“y) (+x , โ€“y)

sin : cos : tan :

+++

sin : cos : tan :

+โ€“ โ€“

sin : cos : tan :

โ€“โ€“ +

sin : cos : tan :

โ€“+ โ€“

(cos, sin)

ALLStudents

Take Calculus

tan๐œƒ=si n ๐œƒcos๐œƒ

2

32

2

Page 14: Warm-Up 3/ Find the measure of

EVALUATING TRIG FUNCTIONS AT ANY ANGLE

1.Find the reference angle.2.Find the corresponding trig value.3.Determine the sign of the angle.

x

y

O

โ€™

Page 15: Warm-Up 3/ Find the measure of

12โˆš32โˆš33

12

โˆš32

โˆš3

โˆš22โˆš22

1

Page 16: Warm-Up 3/ Find the measure of

Example 4: Find the exact value of each expression.

a. cos (โ€“ 240)

x

y

โ€™= 60

= โ€“ cos 60 ยฟโˆ’12

Page 17: Warm-Up 3/ Find the measure of

Checkpoints:1. Find the exact values of the three basic trig functions of given (โ€“ 4 , 3).

r = 5,

2. Find the exact value of the trig function .

r = 1,

3. Sketch the angle and then find the reference angle of 300.

4. Find the exact value of the trig function .

Quadrant III tan is positive.

Page 18: Warm-Up 3/ Find the measure of

Assignment4-3 Worksheet 1-19 all

Page 19: Warm-Up 3/ Find the measure of

Warm-Up 3/28

1. Find the exact value of the trig function .

r = 1,

2. Find the exact value of the trig function .

Quadrant II cos is negative.

Page 20: Warm-Up 3/ Find the measure of

Assignment4-3 Worksheet 1-19 all