Grade 12 Trigonometry

Trig Definitions

Grade 12 Trigonometry

Trig Definitions

Radian Measure•Recall, in the trigonometry

powerpoint, I said that Rad is Bad.

•We will finally learn what a Radian is and how it compares to a degree.

Radian Measure

•One Radian is defined as the measure of an angle that, if placed with the vertex at the center of the circle, intersects an arc of length equal to the radius of the circle.

• One radian measure where the radius and the arc length are the same length.

• Therefore, in this diagram r =b

The Circle and the Radian

• The circumference of a circle.

C =2πr

Circumference of a Unit Circle

What is the circumference of a unit circle where the

radius = 1?

Circumference of the Unit Circle

If the circumference of a unit circle is r =1

C =2π(1)C =2π

Relationship between Degrees

and RadiansIf the circumference of a unit circle is and if a circle has 360 degrees what is the equivalent radian measure for the

following:

2π

90 degrees = radians180 degrees = radians270 degrees = radians360 degrees = radians

Answers 90 degrees = radians

180 degrees = radians

270 degrees = radians3π2

ππ2

Converting from Degrees to

RadiansUnit Analysis:

Recall if you want to cancel degrees on top you must

have degrees on the bottom to cancel the units. Think of

the degree symbol as a ‘unit’ that needs.

1o =

π180o

Convert Degrees to Radians

Convert 210 degrees to Radian Measure using the ratio provided on the previous slide.

210o =210 ×π

180o radians

         =7π6

radians

Warning: Do not use decimals to simplify!!

Warning: Do not use decimals to simplify!!

Convert the following degree measures to radian measures:

30 degrees45 degrees60 degrees90 degrees

120 degrees135 degrees150 degrees180 degrees270 degrees360 degrees

Warning: Do not use decimals to simplify!!

Convert the following degree measures to radian measures:

30 degrees

45 degrees

60 degrees

90 degrees

120 degrees

135 degrees

150 degrees

180 degrees

270 degrees

360 degrees

π6π4π3

π2

2π3

3π45π6π3π2

2π

Converting from Radians to Degrees

1 radian =

180o

π

Think about unit analysis again in this

conversion. If you need to convert to degrees

you need to have radians on the bottom and degrees left on the

top.

Convert Radians to Degrees

6π

6π radians=6π ×180o

π                   =6 ×180o

                   =1080o

Arc Length

To calculate arc length use the formula:

s =rΘ

Arc Length Example

• Determine the radius of a circle in which a central angle of 3 radians subtends an arc of length 30 cm.

s =rΘ30 =r ×10∴ r =3

Coterminal Angles

• Two angles in standard position are coterminal if they have the same terminal side. There are infinite number of angles coterminal with a given angle.

• To find an angle coterminal with a given angle, add or subtract

• For example,

2ππ + 2π = 3π

Angles• A trigonometric angle is determined by rotating a

ray about is endpoint, called the vertex of the angle

• The starting position of the ray is called the initial side and the ending position is the terminal side

Initial Side

Terminal Side

Initial and Terminal Sides

Which are is the initial side and which are is the terminal side?

Angle Direction• If the displacement of the ray from

its starting position is in the counter clockwise position it is assigned a positive measure

• If the displacement of the ray from its starting position is in the clockwise position it is assigned a negative measure

Standard Position•An angle is in standard position

in a Cartesian Coordinate system if its vertex is at the origin and it initial side is the positive x-axis.

Standard Position Graph