Radian Measure and applications Chapter 2 Circular Functions and Trigonometry

A radian An Intro An IntroAn Intro Look at this diagram- applet: Look at this diagram- applet:

What is a radian? The measure of the CENTRAL angle subtended by an arc EQUAL IN LENGTH to the radius of a circle. The measure of the CENTRAL angle subtended by an arc EQUAL IN LENGTH to the radius of a circle. Here is a radian. Here is a radian. (the green bit) (the green bit) There are 2π radians in a circle (find out why later!) There are 2π radians in a circle (find out why later!) Click HERE to see how you get a radian.. There are also some questions to test you/ Click HERE to see how you get a radian.. There are also some questions to test you/HERE

1 Radian 1 Radian

A radian Definition: if the circumference of a circle is 2пr how many r’s will go around the circle? Definition: if the circumference of a circle is 2пr how many r’s will go around the circle?

The relationship Yes 2  is equal to one full turn of a circle Yes 2  is equal to one full turn of a circle So 2  = So 2  = Or it is easier to remember Or it is easier to remember  =  = 180 0

Complete this table RadiansDegrees 3  /4 c  /2 c 90  /4 c 3  /2 c 1 c

Common angles Click here Click here

Arc length Do you want to see the proof or just the formula? Do you want to see the proof or just the formula? Length of an arc l = θ Length of an arc l = θ Circumference 2п Circumference 2п The proportions are the same! The proportions are the same! Since the circumference is 2пr Since the circumference is 2пr Then l = rθ Then l = rθ Remember the angle θ is in radians! Remember the angle θ is in radians!

Area of a sector Area of sector A = θ Area of sector A = θ area of circle 2п area of circle 2п Since the area of a circle is пr 2 Since the area of a circle is пr 2 A = ½ r 2 θ A = ½ r 2 θ Please Remember to use RADIANS Please Remember to use RADIANS

Arc length, area of a sector Area of a sector proof: Area of a sector proof: Arc length: Arc length: Please remember these angles in these two formulae are all in RADIANS! Please remember these angles in these two formulae are all in RADIANS!

Arc length To find the arc length of a circle when the angle subtended is given in radians we use this formula: To find the arc length of a circle when the angle subtended is given in radians we use this formula: Arc length = r  where r is the radius and  is the angle subtended in radians Arc length = r  where r is the radius and  is the angle subtended in radians r

Radian Click here for a game Click here for a game Match here Match here Radian practice with trig functions here Radian practice with trig functions here

Area of a segment Remember another formula or remember the method using common sense? Remember another formula or remember the method using common sense? Let’s use our common sense! Let’s use our common sense! What steps would you have to take to find the area of the segment? What steps would you have to take to find the area of the segment?

Finding the Area of a Segment Please use Radians!

How do I find the area of a segment? Look at the following diagram and follow the hint steps to find the area of the segment. Look at the following diagram and follow the hint steps to find the area of the segment. Shade in the segment in the circle below. Label the triangle AOB. Angle AOB = 0.5 radians and the radius is 12 cm. Shade in the segment in the circle below. Label the triangle AOB. Angle AOB = 0.5 radians and the radius is 12 cm. Find the area of the sector Find the area of the sector Find the area of the triangle using ½ absinC Find the area of the triangle using ½ absinC What is the area of the segment? What is the area of the segment?