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Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions?. Do Now:. A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 270 0. Q II. Quadrant I. terminal side. 90 <  < 180. 0 <  < 90. terminal side.

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Aim what good is the unit circle and how does it help us to understand the trigonometric functions
Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions?

Do Now:

A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 2700.


Angles in standard position

Q II to understand the Trigonometric Functions?

Quadrant I

terminal side

90 <  < 180

0 <  < 90

terminal side

initial side

t.s.

Q III

Q IV

t.s

180 <  < 270

270 <  < 360

  • An angle on the coordinate plane is in standard position when its vertex is at the origin and its initial side coincides with the nonnegative ray of the x-axis.

Angles in Standard Position

y

x

  • An angle formed by a counterclockwise rotation

  • has a positive measure.

  • Angles whose terminal side lies on one of the axes is

  • a quadrantal angle. i.e. 900, 1800, 2700, 3600, 4500 etc.


Co terminal and negative angles

Q II to understand the Trigonometric Functions?

Quadrant I

90 <  < 180

0 <  < 90

initial side

-

t.s.

Q III

Q IV

180 <  < 270

270 <  < 360

Co-terminal and Negative Angles

y

3000 =

x

600

  • An angle formed by a clockwise rotation has a

  • negative measure

  • Angles in standard position having the same

  • terminal side are co-terminal angles.


Angles greater than 360 0

Q II to understand the Trigonometric Functions?

Quadrant I

90 <  < 180

0 <  < 90

Q III

Q IV

180 <  < 270

270 <  < 360

  • Angles whose terminal side rotates more than one

  • revolution form angles with measures greater

  • than 3600.

Angles Greater than 3600

y

4850

1250

x

  • To find angles co-terminal with an another angle

  • add or subtract 3600.

1250 and 4850 are co-terminal


Model problems
Model Problems to understand the Trigonometric Functions?

  • Find the measure of an angle between 00 and 3600 co-terminal with

  • 3850 b) 5750 c) -4050

  • In which quadrant or on which axis, does the terminal side of each angle lie?

  • a) 1500b) 5400 c) -600

215o

315o

25o

x-axis

QIV

QII


Unit circle

hypotenuse to understand the Trigonometric Functions?

side opp.

cos 

side adj.

Unit Circle

y

1

radius = 1

center at (0,0)

cos , sin 

(x,y)

x

-1

1

-1


Aim what good is the unit circle and how does it help us to understand the trigonometric functions1
Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions?

Do Now:

Find the measure of an angle between 00 and 3600 co-terminal with an angle whose measure is -1250.


Value of sine cosine quadrant i

3 to understand the Trigonometric Functions?

Hypotenuse = 2  shorter leg

Longer leg =  shorter leg

Value of Sine & Cosine: Quadrant I

y

1

radius = 1

center at (0,0)

cos 600, sin 600

(x,y)

600

x

-1

1

What is the value of coordinates (x,y)?

300-600-900 triangle

Sine and Cosine values for angles in

Quadrant I are positive.


Value of sine cosine quadrant ii

( to understand the Trigonometric Functions?x,y)

1200

side adj.

3

Hypotenuse = 2  shorter leg

Longer leg =  shorter leg

Sine values for angles in Quadrant II are positive.

Value of Sine & Cosine: Quadrant II

y

1

cos 1200, sin 1200

What is the value

of coordinates (x,y)?

1

60º is the

reference

angle

(180º-120º)

600

x

-1

1

directed distance

A reference angle for any angle in standard

position is an acute angle formed by the terminal

side of the given angle and the x-axis.

What is the cosine/sine of a 1200 angle?

300-600-900 triangle

Cosine values for angles in Quadrant II are negative.


Value of sine cosine quadrant iii

side opp. to understand the Trigonometric Functions?

side adj.

Value of Sine & Cosine: Quadrant III

y

1

What is the value

of coordinates (x,y)?

What is the cosine/sine

of a 2400 angle?

2400

directed distance

60º is the

reference

angle

(240º-180º)

x

-1

600

1

directed dist.

1

(x,y)

cos 2400, sin 2400

Sine and Cosine values for angles in

Quadrant III are both negative.


Value of sine cosine quadrant iv

side opp. to understand the Trigonometric Functions?

(x,y)

Sine values for angles in Quadrant IV are negative.

Value of Sine & Cosine: Quadrant IV

y

1

What is the value

of coordinates (x,y)?

What is the cosine/sine

of a 3000 angle?

60º is the

reference

angle

(360º-300º)

3000

x

-1

600

1

directed dist.

1

cos 3000, sin 3000

Cosine values for angles in Quadrant IV are positive.


Unit circle 12 equal arcs
Unit Circle – 12 Equal Arcs to understand the Trigonometric Functions?


Unit circle 8 equal arcs

Periodic to understand the Trigonometric Functions?

Unit Circle – 8 Equal Arcs

Negative Angles Identities


Value of sine cosine in coordinate plane

y to understand the Trigonometric Functions?

Quadrant II

Quadrant I

x

Quadrant III

Quadrant IV

Value of Sine & Cosine in Coordinate Plane

cos  is +

sin  is +

cos  is –

sin  is +

cos  is +

sin  is –

cos  is –

sin  is –

for any angle in standard

position is an acute angle formed by the terminal

side of the given angle and the x-axis.

The reference angle:


Model problems1
Model Problems to understand the Trigonometric Functions?

  • Fill in the table

  • Quad. Ref.  sin cos 

  • 2360

  • 870

  • -1600

  • -36

  • 13320

  • -3960


Regents prep

y to understand the Trigonometric Functions?

1

x

-1

1

-1

Regents Prep

On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 240 and find the exact value of sin 240o.


Aim what good is the unit circle and how does it help us to understand the trigonometric functions2
Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions?

Do Now:

Use the unit circle to find:

a. sin 1800 () b. cos 1800


Model problems2

to understand the Trigonometric Functions?

(-1,0)

Model Problems

Use the unit circle to find:

a. sin 1800 () b. cos 1800

(x, y) = (-1, 0)

sin 1800 = y

= 0

cos 1800 = x

= -1

180º - quadrantal angle


sin to understand the Trigonometric Functions?

sin 

cos 

cos  = 1

-1

( 1, tan)

( , )?

Tan 

radius = 1

center at (0,0)

cos , sin 

y

1

(x,y)

tan

1

x

-1

1


Trigonometric values
Trigonometric Values to understand the Trigonometric Functions?

+

+


Trigonometric values a c t s

Quadrant I to understand the Trigonometric Functions?

Q II

90 <  < 180

0 <  < 90

Q III

Q IV

180 <  < 270

270 <  < 360

Trigonometric Values - A C T S

y

S Sine is +

A

All are +

x

T

Tangent is +

C

Cosine is +


Need to knows

1 to understand the Trigonometric Functions?

y

r

y

x

-1

x

1

-1

Reciprocal Functions

Negative Angles Identities

csc  = 1/y

sec  = 1/x

cot  = x/y

denominators  0

Need to Knows

When r = 1

sin  = y

cos  = x

tan  = y/x


Model problems3

y to understand the Trigonometric Functions?

(x,y)

1

1

sin  = y

2

2

450

x

-1

1

cos  = x

In a 450-450-900 triangle, the length of the hypotenuse is times the length of a leg.

-1

(x)

length of hypo. =

Model Problems

Using the unit circle, find

cos 450 (/4)

sin 450

tan 450

450-450-900 triangle

A 450-450-900 triangle is an isosceles right triangle.

therefore x = y

cos  = sin 


Model problems4

y to understand the Trigonometric Functions?

1

1

sin  = y

45º

x

-1

1

cos  = x

=

=

-1

Model Problems

Using the unit

circle, find

cos 45º(/4)

sin 45º

tan 45º

(x,y)

cos 45º = x

tan 45 = 1

sin 45º = y


Trigonometric values for special angles

to understand the Trigonometric Functions?

0

30º

/6

45º

/4

60º

/3

90º

/2

sin 

0

1

cos 

1

0

tan 

0

1

UND.

Trigonometric Values for Special Angles

Why is tan 90º undefined?

What is the slope of a line perpendicular to the x-axis?

= slope


Model problems5

What is the cos 510º (17 to understand the Trigonometric Functions?/6)?

  • cos 30º =

  • cos 510º=

Model Problems

What is the tan 135º (3/4)?

  • 135º is in the 2nd quadrant

  • 45º is reference angle (180 – 135 = 45)

  • tan 45º = 1

  • tangent is negative in 2nd quadrant

  • tan 135º= -1

  • 510º is in the 2nd quadrant

  • (510 – 360 = 150)

  • 30º is reference angle (180 – 150 = 30)

  • cosine is negative in 2nd quadrant

≈ -.866…


Model problems6
Model Problems to understand the Trigonometric Functions?


Model problems7
Model Problems to understand the Trigonometric Functions?

Given:

sin 68o = 0.9272

cos 68o = 0.3746

Find cot 112o

  • -0.3746 B) -2.4751

  • C) -0.404 D) 1.0785

reference angle for 112o is 68o; 112o is in QII; tan and cot are negative in QII

WHAT ELSE DO WE KNOW?


Model problems8
Model Problems to understand the Trigonometric Functions?

Express sin 285º as the function of an angle

whose measure is less than 45º.

What do we know?

285º in IV quadrant

the sine of a IV quadrant

angle is negative

-sin 75º

reference angle for 285º is

(360 – 285) = 75º

> 45º

sine and cosine are co-functions

complement of 75º is 15º

< 45º

sin 285º =

= -cos 15º

-sin 75º


Trig functions using radian measures
Trig Functions Using Radian Measures to understand the Trigonometric Functions?

Algebraically:

Find:

sin (π/3)

remember:

π/3 radians

π/3

60º

sin 60º =

≈ .866…

Using the calculator:

Use the mode key:

change setting from degrees to radians

then hit:

sin

2nd

π

÷

ENTER

3

Display: .8660254083


Un unit circle

y to understand the Trigonometric Functions?

1

1

-1

unit circle

1

x

-1

Un-unit circle

 is any angle in standard position with (x, y) any point on the terminal side of  and

r  1


Model problem

4 to understand the Trigonometric Functions?

r

= 5

3

Model Problem

(-3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of .

Q II


Model problem1

-1 to understand the Trigonometric Functions?

r

= 2

Model Problem

is a point on the terminal side of . Find , the sine, cosine, and tangent of .

Q III


Model problem2
Model Problem to understand the Trigonometric Functions?

Tan  = -5/4 and cos  > 0, find sin  and sec 

When tangent is negative and cosine is positive angle is found in Q IV.


Model problem3
Model Problem to understand the Trigonometric Functions?

The terminal side of  is in quadrant I and lies on the line y = 6x. Find tan ; find .

y = mx + b - slope intercept form of equation

m = slope of line

y = 6x

m = 6 = tan 

Q I


Model problem4
Model Problem to understand the Trigonometric Functions?

The terminal side of  is in quadrant IV and lies on the line 2x + 5y = 0. Find cos .

y = mx + bslope intercept

form of equation

tan  = m = -2/5


Templates

y to understand the Trigonometric Functions?

1

1

45º

x

-1

1

-1

Templates


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