4.1 Day 1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure –Convert between degrees & radians –Draw angles in standard position Pg. 458 #8, 10, (every other even), (all the evens) Remember to follow the directions! - Especially with the evens section.

Angles An angle is formed when two rays have a common endpoint. Standard position: one ray lies along the x-axis extending toward the right Positive angles measure counterclockwise from the x-axis Negative angles measure clockwise from the x-axis

The measure of the central angle in radians can be found with the following formula: 1.A central angle,, in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of ?

Angle Measure Degrees: Full circle = 360 degrees Half-circle = 180 degrees Right angle = 90 degrees Radians: One radian is the measure of the central angle that intercepts an arc equal in length to the length of the radius Full circle = 2 radians Half circle = radians Right angle = radians

How many times can you trace an arc length equal to the size of the radius as you move around the circumference of a circle? Well, here's the formula for the circumference of a circle: C is the circumference, and r is the radius. Looks like there are two pi radii around the circumference of a circle, or about 2 times 3.14, i.e., about 6.28, radii around the circumference of a circle. Therefore, there are 2 times π radians in a full circle. Each one of these radius lengths would designate one radian, so there are about 6.28 radians in a full circle. The following diagram shows this:

Radian Measure Recall half circle = 180 degrees= radians This provides a conversion factor. Since they are equal, their ratio=1. so we can convert from radians to degrees (or vice versa) by multiplying by one of the following ratios: OR Example: Convert 270 degrees to radians

2. Convert each angle from degrees to radians: a.30 o b. 270 o c o 3. Convert each angle from radians to degrees: d. e. f. 6

Let’s learn how to draw angle positions counting in radian units. Note: This is a VERY important skill which must be mastered in this unit. Unit: Unit:

This diagram shows the most commonly utilized angles employed in trigonometry and physics (where radians are much more often used than degrees). Label the equivalent angle measures in degrees at each standard position of the diagram  Being able to make this diagram from scratch and without help will be an essential skill to master in this unit of the precalculus textbook. The sooner you can do this, the better off you will be. (The ? on the diagram is a non-erasable error – please disregard) Your textbook has the fully labeled diagram on Pg. 453

Draw and label each angle: a) b) c) d)

Draw and label each angle: e) f)