A famous artisan located in Changmai, Thailand makes ornate umbrellas and fans for her business.  She has an order for three dozen umbrellas having a diameter of 2 meters.  Bamboo slats that support each circular umbrella divide the umbrella into 8 sections, or 

.  Each section will be covered with a different color fabric.  How much fabric of each color will the umbrella maker need to complete the order?  You will revisit this craftsperson conundrum later in this lesson.

There are many real-world applications that can be solved more easily using angle measurement other than degree.  This other useful unit is called a 

.  A radian is based directly on the concept of the unit circle from previous lessons.  Aspects to remember:  

So, consider an angle θ in standard position whose terminal side contains point 

 at (x , y).  One full rotation around the origin would be measured as 360 in degrees, but now, we can also classify it as a measurement of 2π 

.  An angle measuring 180°, half of the circle, would be π radians, 90° would be π/2 radians and so on.

🔑Given: 1 degree = π/180°.  1 radian = 180°/π.  Convert the following into degrees or radians in terms of π.

a) 210°      b) 11π/3      c) -75°      d) 1250°      e)7π/12      f) 17.5      g) -2π/3

To ease the transition into radians as units, you can use prior knowledge of the unit circle to determine trigonometric values for angle measures.  For example, take a measurement of θ as 4π/3.  The terminal side of this angle is in Quadrant III, so to find a reference angle, use the formula θ - 180°, or, in this case, θ - π.  So, 4π/3 - π leaves you with a reference angle of π/3. 🔑 h) What is the value of cos 4π/3?  Write this and the other two trigonometric ratios in your notebook.

Radian measures can be used to find the length of a 

.  An arc is simply a portion of a circle.  The arc is often defined by the 

 that intercepts it, or any angle whose vertex lies at the center of that circle.  If two central angles in two different circles are congruent, the ratio of the lengths of their intercepted arcs is equal to the ratio of the measures of their radii.  

🔑Use the following angle measurements.  Find the length of its intercepted arc in a circle with a radius of 14 centimeters (cm).  Round to the nearest tenth if necessary.

j) 2π/3          k) 150°          m) 5π/12          n) 3π/11          o) 282°          p) 320° 

 probably has all kinds of associations to it, but in terms of Trigonometry it refers to the region bounded by a central angle and its intercepted arc.  A area of a sector in graph form is denoted by shading the sector.  If the area of a circle with radius 

2, then the area of a sector with central angle θ can be defined with the relationship 

 = ½θr2.  When finding the area of a sector, you can use the entire angle measure of θ and ignore the reference angle.  🔑q) Try it: Find the area of a sector if θ = 5π/6 radians and the radius is 16 cm.  Round to the nearest tenth.

🔑t) Recall the example of umbrella creation from earlier in this lesson.  How much fabric of each color will the umbrella craftswoman need to complete the order of three dozen umbrellas?  Draw a diagram of one umbrella and take into account the different sectors needed to solve.

🔑Find the area of each sector with given central angle θ and the radius.

z) Refer to the image below.  A wheel has a radius of 2 feet (ft).  As it turns, a cable connected to a box winds onto the wheel.  🔑1. How far does the box move if the wheel turns 225° in a counterclockwise direction?  🦄2. Find the number of degrees the wheel must be rotated to move the box 5 ft.

Students will use trigonometric familiarity to translate the unit circle into new pieces and ways of measurement.

A Face for Radian