1. 6-1 Angle Measures The Beginning of Trigonometry

2. New Way of Thinking We were used to drawing figures in an x/y plane and looking at their relationships. Now we are going to base everything on a circle drawn on the x/y plane. We’ll be looking at triangles created and the relationships of the sides of the triangles. This, my friends, is trigonometry!!

3. The Radian The Radian is a new way to measure an angle. We are used to using degrees to indicate the size of an angle. Now we will use this new measuring tool. Its like being in Canada - the kilometer vs. the mile. The radian vs. the degree - either way, it indicates the specific size of an angle.

4. The Radian Definition: One Radian is the measure of a central angle of a circle that is subtended by an arc whose length = r. r r One radian

5. Some Variables commonly used Arc Length = s Central Angle = Radius = r Now we’re going to see how these relate

6. I’m going to draw a picture with a central angle = 3 radians, then find the relationship between arc length, central angle, and radius. r r r r This sector = 3r. Therefore, the central angle = 3 radians. s = ·r Therefore

7. What is the radian measure of a whole circle? For a whole circle, s = circumference = So, What does this mean? We now have a conversion factor (dimensional analysis) or

8. Why do this? Examples: 1. Convert 55  to radians. 2. Convert to degrees.

9. Now what? Do you remember areas of sectors from Geometry? Essentially, a sector is a piece of pizza. The area is dependant on the central angle right? For any particular circle, the radius is constant. Only the central angle changes. So…

10. Note: central angle should be in radians!! What would the formula be if the angle was in degrees? You’d need to include the conversion factor Remember?

11. Examples 3.Find the area of the sector of a circle with radius 4 if the central angle = 120 . 4. Find the area of the sector of a circle with radius 2 if the central angle = 5. Find the area of the sector of a circle with radius 2 if the subtended arc = 6.