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Unit Circle

Unit Circle. Activity. Use your protractor to measure the angles whose terminal sides lie along the hash marks of the unit circle Then, cut out the two triangles at the bottom of the page. Activity.

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Unit Circle

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  1. Unit Circle

  2. Activity • Use your protractor to measure the angles whose terminal sides lie along the hash marks of the unit circle • Then, cut out the two triangles at the bottom of the page.

  3. Activity • The hypotenuse both triangles will represent the radius of the unit circle. So the hypotenuse will equal ______ • For the 45-45-90 triangle, each leg is ______- • For the 30-60-90 triangle, the shorter leg is ____ and the longer leg is ____. Label these side measures as well as the angle measures on both sides of your triangles.

  4. Directions • Find the exact location (x, y) of the hash marks located around the unit circle. You can do this by positioning (i.e. rotating and flipping around) the triangles within the circle. • REMINDERS… • The hypotenuse must be lined up with the hash marks • One leg of the triangle must always remain on the x-axis • Use symmetry to fill out the 2nd, 3rd and 4th quadrants. • Cut out your circle and paste it on construction paper. Keep it in a safe place and always bring it to class.

  5. Conclusions • Every point (x, y) on the unit circle will correspond to an angle Ө. • Since sin Өand cos Өare defined using the unit circle, they are called circular functions. • What are the domain and range of sine and cosine?

  6. Unit Circle Definitions • sin Ө= y • cos Ө= x • tan Ө= y/x • cot Ө= x/y • sec Ө= 1/x • csc Ө = 1/y Since division by 0 is undefined, there are several angle measures that are excluded from the domain of these 4 trig functions

  7. Example • Use the unit circle to find each value: • cot(270°) • sec(90°)

  8. What’s your sign? • What determines the sign of a trig ratio? • The quadrant in which the terminal side of the angle lies of will dictate the sign of the x and y coordinate. • All Students Take Calculus

  9. Example • Use the unit circle to find the values of the 6 trigonometric functions for a 210° angle. • sin 210 = _________ • cos 210 = _________ • tan 210 = _________ • csc 210 = _________ • sec 210 = _________ • cot 210 =_________

  10. Example • Use the unit circle to find the values of the 6 trigonometric functions for a 135° angle. • sin 135 = _________ • cos 135 = _________ • tan 135 = _________ • csc 135 = _________ • sec 135 = _________ • cot 135 = _________

  11. What if your angle is negative? • sin (-240°)? • Add revolutions (360) until you find a coterminal angle between 0 and 360

  12. What if your angle is large? • cos 1050° • Subtract revolutions (360) until you find a coterminal angle between 0 and 360

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