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

The Circle and the Radian The circumference of a circle.

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

Relationship between Degrees and Radians If the circumference of a unit circle is and if a circle has 360 degrees what is the equivalent radian measure for the following: 90 degrees = radians 180 degrees = radians 270 degrees = radians 360 degrees = radians

Answers 90 degrees = radians 180 degrees = radians 270 degrees = radians

Converting from Degrees to Radians Unit 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.

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

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

Converting from Radians to Degrees 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

Arc Length To calculate arc length use the formula:

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

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,

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