7.6 NORMAL FORM OF A LINEAR EQUATIONBy the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

WHAT IS A “NORMAL” LINE We find an equation of a line using a point and

a slope Distance between two points comes from the

Pythagorean theorem

Slope can be found using “Normal” mean perpendicular

Perpendicular slopes are OPPOSITE sign and RECIPROCAL

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

SIDES OF A TRIANGLE - Greek letter can be pronounced either

fee (as in a bank fee) or fi (rhymes with pie)

Thus, the sides of our triangle can be found using the angle and the length of the hypotenuse

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

𝜙𝑦

𝑥

𝑝

WHAT IS A “NORMAL” LINE is the distance between

the origin and the line (distance is measured

PERPENDICULARLY) is the angle made with the

positive x-axis

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

x

y

A

B

M

C

O

p

𝜙

𝜙

𝜙

We want the

equation of THIS line

WHAT IS A “NORMAL” LINE How do we get the normal

line?

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

x

y

A

B

M

C

O

p

𝜙

𝜙

𝜙𝑝 ∙ sin 𝜙

𝑝 ∙cos𝜙

(𝑝 ∙cos𝜙⏟𝑥1

,𝑝 ∙ sin 𝜙⏟𝑦1 )

𝑚=− cos𝜙sin 𝜙

𝑚=sin𝜙cos𝜙

We want the

equation of THIS line

HOW IS NORMAL FORM DIFFERENT THAN STANDARD FORM?Normal Standard

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

Why do you have a problem with the last bullet points?What is the difference between Normal and Standard?

Ratio coefficients

Positive leading coeff.Non ratio coefficients

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH THE POSITIVE X AXIS

A.

B.

𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0

x

yA

x

yB

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH THE POSITIVE X AXIS

C.

D.

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0

x

yC

x

yD

HOW DO WE CONVERT FROM STANDARD FORM TO NORMAL FORM?Normal Standard

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

Divide everything by , use the opposite sign of the value for C

This is a sin value

This is a

cos value

Opposite, y

Hypotenuse, p

𝜙𝑦

𝑥

𝑝

adjacent, x𝜙

𝐵

𝐴

√ 𝐴2+𝐵2

WHERE ARE ALL THE ANGLES?? Which quadrant is being described by each?

What is the measure of an angle? Values like are found on the unit circle, we can

give EXACT angles Ratios that we don’t recognize can still be found

using

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM

A. What do we divide by? How do we know which sign to use?opposite of C

??? WHAT??? This is not in quadrant 3??The calculator gives you the PRINCIPAL value, you need to translate that to the appropriate quadrant

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

are not fractions

For the HW this is

acceptable, UNLESS it’s a

unit circle value

What quadrant is this angle in? 𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

What quadrant is this angle in? If you can identify

the angle at this

step from sin and

cos, you may do so.

EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM

B. What do we divide by? How do we know which sign to use?opposite of C

Which of these will give use the quadrant we want?

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

are not fractions

𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0For the HW

this is acceptable,

UNLESS it’s a unit circle

value

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM

C. What do we divide by? How do we know which sign to use?opposite of C

This angle is in quadrant 4 so

What if our angle was in quadrant 2?Use as the reference angle for QII

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

are not fractions

For the HW this is

acceptable, UNLESS it’s a

unit circle value

What quadrant is this angle in? 𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM

What do we divide by?

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

are not fractions

10√2

=10 √22

=5√2

EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM

What do we divide by? How do we know which sign to use?opposite of C+

Which of these will give use the quadrant we want?

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

are not fractions

10√2

=10 √22

=5√2

What quadrant is this angle in? 𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0

For the HW this is acceptable, UNLESS it’s a

unit circle value

SUMMARY1. Write the equation in normal form.2. Write the standard form of the equation of a

line for which the length of the normal is and makes and angle of with the positive .

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.

SUMMARY1. Write the equation in normal form.

2. Write the standard form of the equation of a line for which the length of the normal is and makes and angle of with the positive .

By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.