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Algebra II Honors—Day 74

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Reminders No food/drinks/electronics. Put them all away NOW. Take-Home Test #8 Due Tuesday, May 6 Essential Question/New Material

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

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Review of Right Triangle Trigonometry From Geometry you learned: Opposite (across from the angle) Adjacent (beside the angle) Hypotenuse (longest side) In Geometry, represented an acute angle in a right triangle. In this unit, we’ll extend this idea to let it include ALL angles.

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Review of Right Triangle Trigonometry From Geometry you also learned: – No matter what the size of the triangle, the trig functions for that angle are ALWAYS the same ratio. So, no matter what, for example

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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.

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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)

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Unit Circle/Angle Measures Since the circumference of a circle is and the circumference of the unit circle is, there are radians in a circle. Therefore, radians is equal to 360⁰.

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Unit Circle/Angle Measures MEMORIZE 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 http://teachers.henrico.k12.va.us/math/ito_08/Pics/UnitCircle.png

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Unit Circle/Angle Measures Additional Notes Negative angle measures start from the positive x-axis but turn CLOCKWISE. Angle measures greater than 360 ⁰ (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.

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Unit Circle/Angle Measures Additional 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)

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http://aventalearning.com/content168staging/2008Trigon ometry/unit3/images/MTH08-68.18243.jpg http://www.regentsprep.org/regents/math/algtrig/AT T3/standardangle.gif http://images.tutorcircle.com/cms/ images/tcimages/abc.gif Angles between 0 and 2π are in “standard position.”

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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)

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Homework MEMORIZE THE UNIT CIRCLE and complete the problems on the sheet—Quiz next class period! Work on Take-Home Test

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And because we are dealing with the unit circle here, we can say that for this special case, Remember:

And because we are dealing with the unit circle here, we can say that for this special case, Remember:

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