14 Trigonometry 1: Standard Position and Radian Measure Model 1: Standard Position for Angles ictured below are several points on the Unit Circle (a circle centered at the origin with radius 1), measured according to their angles in standard position. A 120 2170 310 a B C F O a / 419 -120 -240 F 1. Angle measurements always start on the - axis. positive / negative c/ y 2. How are positive and negative angles measured differently? 3. Each point has an angle labeled a with it, known as the reference angle for that point. The reference angle is always . and is always measured to the - axis. acute / obtuse Ily 4. Compute the reference angle for each point. Fill in these blanks to help you get started. Full Circle : - Half Circle : _ One Quadrant : Reference angles: A : B : C : D : _ E : F : _ Note: The reference angle is always positive, even if the original angle is negative.5. Two of the points in Model I are in the exact same place on the circle. Which ones? and 6. When two different angles lead to the same point on the circle, those angles are called coterminal. Find the difference between the angles (i.e., subtract one from the other) for the points you identified in #5. 7. What portion of a circle does your answer to #6 represent? 8. Find two more angles (different from those in Model 1), one positive and one negative, that are coterminal with the two you identified in #5. 9. For each angle below, . sketch the angle in standard position . compute and label the reference angle . find two coterminal angles (one positive and one negative) (a) 2550 (b) 8750 (c) -323 (d) -500 10. Complete this sentence: of The difference between any pair of coterminal angles must be aModel 2: Radian Measure In Calculus, measuring angles with degrees becomes problematic, because the number 360 for a full circle is arbitrary. It is better to measure angles in a way that directly associates them with distance. 0 = 1 radian 0 = 2 radians 0 = 3 radians Definition: The radian measure of an angle is equal to the length of the arc it creates on the Unit Circle. 11. Complete these sentences, which refer to the accompanying picture: Since a angle represents one full circle, the length of the arc it creates is equal to the of the circle. Since the radius of the Unit Circle is -, the length of this arc of the Unit Circle is 12. The angles below, whose points fall precisely on an axis, are known as quadrantal angles. Find the exact values (not approximations) for their radian measures. . . . . .. 0 = _ - radians #11 0 = - radians 0 = radians 0 = radians 13. In addition to the quadrantal angles, the most important angles to be familiar with are 30, 45, and 60. Convert each to radians. 85Hint: What fraction of 180 is each angle? 30 = radians 450 = radians 60 = radians Note: Though we have done so to this point, typically we do not include the word "radians" with our angle measures. That is, we assume that all angles are measured in radians unless the degree symbol is present. 14. Rewrite your answer to #10 using radians instead of degrees: The difference between any pair of coterminal angles must be a of 15. For each angle below, . sketch the angle in standard position . compute and label the reference angle . find two coterminal angles (one positive and one negative) You should perform all of your calculations entirely in radians. That is, do not convert to degrees, solve the problem, then convert your answer back to radians. (a) - . T (b) - . (c) 12 . T (d) -. T15 Trigonometry 2: Special Angles Model 1: Coordinates for 450 icutred below is a portion of the Unit Circle with a triangle drawn within it. P = b 45 a 1. What is the length of the radius of this circle? 2. What is the length c? 3. Which of the triangle's side lengths represents the x-coordinate of the point P? 4. Which of the triangle's side lengths represents the y-coordinate of the point P? 5. What is the measure of the angle ? Explain. 6. Based on your answer to #5, what is the relationship between the lengths a and b? 7. Use the relationship you found in #6 along with the Pythagorean theorem to determine the values of a and b. Then, fill in the appropriate values for the coordinates of P. Note: Stop here and check in with your instructor or a TA before continuing. 87triangle is drawn Model 2: Coordinates for 60 _ . ' of the red below is a portion of the Unit Circle with a triangle drawn within it. An identical COPY f | Plant next to it, reected across its vertical side. 3. What is the length c? 9. What is the measure of the angle ,3? t? Explain your 10. Consider the main triangle and its copy together as a single, large triangle. What type of triangle iS i answer. i/Jr 11. Based on your answer to #10, determine the length (1. Explain your answer. 12. Use the Pythagorean theorem to determine the value of b, then ll in the appropriate coordinates in the picture. 99 13. A copy of the triangle from Model 2 is drawn below. Fill in its angles and side lengths, then fill in the coordinates for 30 in the picture below. 30 Model 3: First quadrant special angles Fill in the coordinates for all three special angles in the picture below. Also, convert the measure of each angle to radians. 60% = 450 = C 30% = Note: Stop here and check in with your instructor or a TA before continuing