Presentation on theme: "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."— Presentation transcript:
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: 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.
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
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 is and the circumference of the unit circle is, there are radians in a circle. Therefore, radians is equal to 360⁰.
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
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.
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)
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.”
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