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# Unit Circle

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by

Tweet## Jessica Lindow

on 13 September 2012#### Transcript of Unit Circle

And the Six Trigonometric Values The Unit Circle Review Special Triangles from last time Hint: There should be one in

each of the four quadrants Take your red colored pencil and

draw in the four 30 degree triangles It should look something like this 45° 45° ? 45° 45° 45° 45° x ? x x ? 45° 45° x x √2 x 30° 60° ? 30° 60° x 30° 60° x 30° 60° x ? √3 x ? √3 x 2x 30° 60° 30° 60° 30° 60° x √3 x 2x Now take

and plug in

x=1 30° 60° 1 √3 2 Next we want

the hypotenuse to

be 1, so divide everything by 2 30° 60° 1/2 √3/2 1 To get: What does our example tell us about our triangle on our UNIT circle? 30° 60° 1 1/2 √3/2 Similarily, by knowing what signs go in what quadrant we obtain: Sin(30°)= ? Cos(30°)= ? Tan(30°)= ? 30° 60° 1 1/2 √3/2 Sin(30°)= Y= 1/2 Cos(30°)= X= √3/2 Tan(30°)= Y/X= √3 30° 60° 1 1/2 √3/2 1 1 1 1/2 -1/2 -1/2 -√3/2 Hint: There should be one in

each of the four quadrants Take your green colored pencil and

draw in the four 45 degree triangles . It should look something like this 45° 45° Hint: There should be one in

each of the four quadrants Take your blue colored pencil and

draw in the four 60 degree triangles It should look something like this 60° 30° 45° 45° x x √2 x Again, plug

in x=1 45° 45° 1 1 √2 To make

the hypotenuse one divide everything

by √2 45° 45° 1/√2 1/√2 1 To get : What does our example tell

us about our unit circle? 45° 45° Sin(45°)= ?

Cos(45°)= ?

Tan(45°)= ? 45° 45° 45° 45° 1 1/√2 1/√2 Sin(45°)= Y= 1/√2

Cos(45°)= X= 1/√2

Tan(45°)= Y/X= 1 1 1/√2 1/√2 45° 45° Recognizing signs

again we get: 1 1/√2 1/√2 1/√2 -1/√2 -1/√2 -1/√2 1 1 1 Notice we are already

back to a 30 degree

triangle tipped on it's side 60° 30° So filling in our sides

of the triangle

we get: 60° 30° √3/2 -√3/2 -√3/2 √3/2 -1/2 1/2 1 1 1 1 Important Formulas to the UNIT circle: Sin(θ)= Opp./Hyp. = Y/1= Y Cos(θ)= Adj./Hyp.= X/1= X Tan(θ)= Sin(θ)/ Cos(θ)=Y/X Now you can label

the rest of your graph Other Important

Formulas to Note: Csc(θ)= 1/Sin(θ) Sec(θ)=1/Cos(θ) Cot(θ)=1/Tan(θ) Example: If: Sin(30°)= 1/2

Cos(30°)= √3/2

Tan(30°)= √3 What do:

Csc(30°)=?

Sec(30°)=?

Cot(30°)=? Csc(30°)=1/(1/2) =2

Sec(30°)= 1/(√3/2)= 2/√3

Cot(30°)= 1/√3 Homework: Fill out the table at the bottom of page three using the graphs and examples we did in class

DUE NEXT CLASS (X,Y)= (Cos(θ), Sin(θ)) Helpful Hint: Real World Applications Converting Degrees to Radians: Multiply degrees by (π/180°) Ex: 30° to Radians 30°(π/180°)= π/6 Becomes: So the unit circle can also be represented in Radians like this:

Full transcripteach of the four quadrants Take your red colored pencil and

draw in the four 30 degree triangles It should look something like this 45° 45° ? 45° 45° 45° 45° x ? x x ? 45° 45° x x √2 x 30° 60° ? 30° 60° x 30° 60° x 30° 60° x ? √3 x ? √3 x 2x 30° 60° 30° 60° 30° 60° x √3 x 2x Now take

and plug in

x=1 30° 60° 1 √3 2 Next we want

the hypotenuse to

be 1, so divide everything by 2 30° 60° 1/2 √3/2 1 To get: What does our example tell us about our triangle on our UNIT circle? 30° 60° 1 1/2 √3/2 Similarily, by knowing what signs go in what quadrant we obtain: Sin(30°)= ? Cos(30°)= ? Tan(30°)= ? 30° 60° 1 1/2 √3/2 Sin(30°)= Y= 1/2 Cos(30°)= X= √3/2 Tan(30°)= Y/X= √3 30° 60° 1 1/2 √3/2 1 1 1 1/2 -1/2 -1/2 -√3/2 Hint: There should be one in

each of the four quadrants Take your green colored pencil and

draw in the four 45 degree triangles . It should look something like this 45° 45° Hint: There should be one in

each of the four quadrants Take your blue colored pencil and

draw in the four 60 degree triangles It should look something like this 60° 30° 45° 45° x x √2 x Again, plug

in x=1 45° 45° 1 1 √2 To make

the hypotenuse one divide everything

by √2 45° 45° 1/√2 1/√2 1 To get : What does our example tell

us about our unit circle? 45° 45° Sin(45°)= ?

Cos(45°)= ?

Tan(45°)= ? 45° 45° 45° 45° 1 1/√2 1/√2 Sin(45°)= Y= 1/√2

Cos(45°)= X= 1/√2

Tan(45°)= Y/X= 1 1 1/√2 1/√2 45° 45° Recognizing signs

again we get: 1 1/√2 1/√2 1/√2 -1/√2 -1/√2 -1/√2 1 1 1 Notice we are already

back to a 30 degree

triangle tipped on it's side 60° 30° So filling in our sides

of the triangle

we get: 60° 30° √3/2 -√3/2 -√3/2 √3/2 -1/2 1/2 1 1 1 1 Important Formulas to the UNIT circle: Sin(θ)= Opp./Hyp. = Y/1= Y Cos(θ)= Adj./Hyp.= X/1= X Tan(θ)= Sin(θ)/ Cos(θ)=Y/X Now you can label

the rest of your graph Other Important

Formulas to Note: Csc(θ)= 1/Sin(θ) Sec(θ)=1/Cos(θ) Cot(θ)=1/Tan(θ) Example: If: Sin(30°)= 1/2

Cos(30°)= √3/2

Tan(30°)= √3 What do:

Csc(30°)=?

Sec(30°)=?

Cot(30°)=? Csc(30°)=1/(1/2) =2

Sec(30°)= 1/(√3/2)= 2/√3

Cot(30°)= 1/√3 Homework: Fill out the table at the bottom of page three using the graphs and examples we did in class

DUE NEXT CLASS (X,Y)= (Cos(θ), Sin(θ)) Helpful Hint: Real World Applications Converting Degrees to Radians: Multiply degrees by (π/180°) Ex: 30° to Radians 30°(π/180°)= π/6 Becomes: So the unit circle can also be represented in Radians like this: