Algebra II Honors—Day 74

Reminders

• No food/drinks/electronics. Put them all away NOW.

• Take-Home Test #8 Due Tuesday, May 6

• Essential Question/New Material

Essential Questions

• What is a radian, and how do I use it to determine angle measure on a circle?

Review of Right Triangle Trigonometry

• From Geometry you learned:

cos

sintan

cos

sin

adj

opp

hyp

adj

hyp

opp

Opposite (across from the angle)

Adjacent (beside the angle)H

ypot

enus

e (lo

nges

t sid

e)

In Geometry, represented an acute angle in a right triangle. In this unit, we’ll extend this idea to let it include ALL angles.

Angle Measures

• Usually in Geometry, angles are measured in degrees.– A circle has 360⁰– A half-circle has 180⁰– A right angle measures 90⁰

• An alternate way to measure angles in a circle is in a measure called radians.

Unit Circle/Angle Measures

• A unit circle is a circle centered at the origin with a radius of 1 unit.

• In a unit circle, a radian is defined as the measure of an angle whose rays intersect an arc length of 1 unit.

• In this diagram, the measure of the angle is one radian.

Arc length=1 unit (same as the radius of the circle)

Unit Circle/Angle Measures

• Since the circumference of a circle isand the circumference of the unit circle is , there are radians in a circle.

• Therefore, radians is equal to 360⁰.

rC 2

2C 2

2

Unit Circle/Angle MeasuresMEMORIZE OR BE ABLE TO FIGURE OUT

Angle measures begin from the positive x-axis.

Positive angle measures turn counter-clockwise.

To convert:degrees to radians multiply by

radians to degrees multiply by

180

180http://teachers.henrico.k12.va.us/math/ito_08/Pics/UnitCircle.png

Unit Circle/Angle MeasuresAdditional Notes

Negative angle measures start from the positive x-axis but turn CLOCKWISE.

Angle measures greater than 360⁰ (2π radians) or less than 0⁰ (0 radians) have “coterminal angles” that fall between 0⁰ and 360⁰ (or between 0 and 2π radians) in standard position.

Unit Circle/Angle MeasuresAdditional Notes

Example: an angle of 400⁰ would end up at the same location as a 40⁰ angle (so 400⁰ and 40⁰ are coterminal)

(subtract a multiple of 360 for degrees or a multiple of 2π for radians)

Example: an angle of –π/2 radians would end up at the same point as an angle of 3π/2 radians (add a multiple of 360 for degrees or a multiple of 2π for radians)

http://aventalearning.com/content168staging/2008Trigonometry/unit3/images/MTH08-68.18243.jpg

http://www.regentsprep.org/regents/math/algtrig/ATT3/standardangle.gif

http://images.tutorcircle.com/cms/images/tcimages/abc.gif

Angles between 0 and 2π are in “standard position.”

20 3600

Graded Classwork

• With a partner or on your own, complete the Angles and Angle Measure handout. Turn in at the end of the period for a grade (one sheet for each pair of students)

Homework

• MEMORIZE THE UNIT CIRCLE and complete the problems on the sheet—Quiz next class period!

• Work on Take-Home Test