Angles and the Unit Circle

An angle is in standard position when: 1) The vertex is at the origin. 2) One leg is on the positive x – axis. (This is the initial side.) 3) The second ray moves in the direction of the angle (This is the terminal side.)

If the movement from the initial side to the terminal side of the angle is counterclockwise, then the angle measures positive °

If the movement from the initial side to the terminal side of the angle is clockwise, then the angle measures negative. – 225 °

–315 ° 240 ° –110 ° 1) 2)3)

Two angles in standard position that have the same terminal side are coterminal angles. To find a coterminal angle between 0 ° and 360 ° either add or subtract 360 ° until you get the number that you want. Find the measure of an angle between 0 ° and 360 ° coterminal with each given angle: 4) 575 ° 215 ° 5) –356 ° 4 ° 6) –210 ° 150 ° 7) –180 ° 180 °

The Unit Circle: 1) Is centered at the origin, 2) Has a radius of 1, 3) Has points that relate to periodic functions. Normally, the angle measurement is referred to as θ (theta). 1 1

 For all values using SOH, CAH, TOA the H value is always 1.  We can use the Pythagorean Theorem to find the rest.  cos θ is the x coordinate.  sin θ is the y coordinate.  Let’s find sin (60°) and cos (60°).  On a triangle the short side is ½ the hypotenuse.  So, cos (60°) = ½.  a 2 + b 2 = c 2 (½) 2 + b 2 = 1 2 ¼ + b 2 = 1 b 2 = ¾ b = √(¾) = √(3)/2  So, sin (60 ° ) = √(3)/2 1 ½ √3 2

 Continue to find the values on the Unit Circle  Find cos 0° and sin 0°  Find cos 30° and sin 30°  Find cos 45° and sin 45°  Find cos 90° and sin 90° 1 1 (1, 0) (0, 1) (½, √3 / 2 ) ( √2 / 2, √2 / 2 ) ( √3 / 2, ½)

 These patterns repeat for the right x and y values.  The values can be either positive or negative based on the x and y axes.  Use this information to fill in the worksheets with exact values

Locate the Unit Circle diagram from before. 8)9) sin (–60°) = –√(3)/2 cos (–60°) = ½ sin (–60°) = –½ cos (–60°) = √(3)/2 10) –390° 11) –30°

 During this lesson we completed page 708 # 1 – 27 odd.  For more practice, complete the even problems