Unit Introduction to Trigonometr The Unit Circle Unit.) William Bill) Finch Mathematics Department Denton High School Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Lesson Goals When ou have completed this lesson ou will: Find values of trigonometric functions for an angle. Find the values of trigonometric functions using the unit circle. Unit Circle / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Trigonometric Functions of An Angle is an angle in standard position, P, ) is a point on the terminal side, and r is the distance from P to the origin denominators 0): P, ) r sin = r cos = r tan = csc = r sec = r cot = Recall the equation of a circle centered at the origin: r = + r = + Unit Circle / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample 1 The point, ) is on the terminal side of an angle in standard position. Find the eact values of the si trigonometric functions of. Unit Circle / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Quadrental Angles Quadrental angles terminate on an ais. r, 0) 0, r) r, 0) 0, r) = 0 or 0 radians = 90 or / radians = 180 or radians = 70 or / radians Unit Circle 5 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample Find the eact value of each trigonometric function, if defined. If not defined, write undefined. a) cos b) tan 70 ) c) sec d) sin 5 Unit Circle / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Angles Not Acute or Quadrental Quadrant II a, b) b r a sin = b r cos = a r sin = b r cos = a r tan = b a tan = b a Unit Circle 7 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Angles Not Acute or Quadrental Quadrant III b a, b) a r sin = b r cos = a r tan = b a sin = b r cos = a r tan = b a Unit Circle 8 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Angles Not Acute or Quadrental Quadrant IV a r b a, b) sin = b r cos = a r tan = b a sin = b r cos = a r tan = b a Unit Circle 9 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Reference Angles If is an angle in standard position, its reference angle is the acute angle formed b the terminal side of and the -ais. = = 180 = = 180 = = 0 = Unit Circle 10 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample Sketch the angle and the identif its reference angle. a) 150 b) 15 c) d) 5 Unit Circle 11 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Evaluating Trigonometric Functions of An Angle 1. Sketch the angle.. Determine the reference angle.. Find the value of the trig function for.. Determine the sign pos or neg) based on the quadrant containing the terminal side of. Quad II sin : + cos : tan : Quad III sin : cos : tan : + Quad I sin : + cos : + tan : + Quad IV sin : cos : + tan : Unit Circle 1 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Special Reference Angles radians) degrees) 0 5 0 sin cos tan 1 1 1 Unit Circle 1 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample Find the eact value of each epression. a) sin b) sec 15 c) tan 150 d) cos 10 ) Unit Circle 1 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample 5 9 Let sec =, where sin > 0. Find the eact values of 5 the remaining five trigonometric functions of. Unit Circle 15 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar The Unit Circle A unit circle is a circle of radius 1 centered at the origin. The radian measure of a central angle is = s r = s 1 = s 1, 0) 0, 1) r r s 1, 0) so the arc length intercepted b equals the angle s radian measure. 0, 1) Unit Circle 1 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar The Unit Circle and the Wrapping Function Place a number line verticall tangent to a unit circle at 1, 0). Wrap this line around the circle counterclockwise for positive values and clockwise for negative values), each point t on the line would map to a unique point P, ) on the circle. This is referred to as the wrapping function wt). Since r = 1, the si trigonometric rations of angle t can be defined in terms of just and. P, ) t t 1 1, 0) Unit Circle 17 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Trigonometric Functions on the Unit Circle sin t = cos t = csc t = 1 sec t = 1 P, ) Pcos t, sin t) t 1 t tan t = cot t = And, of course, no denominator = 0. These functions are referred to as circular functions Unit Circle 18 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar 1-Point Unit Circle 1, ), ) 10 ), 1 15 1, 0) 180 150 5 0, 1) 90 ) 1 ) 0 ) 5, 1 0 0 0 1, 0) 7 11 5 7 10 5 0 5 ), 1 15 ), 1 0 00, ) 70 1, ) 1, ), ) 0, 1) Unit Circle 19 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar 1-Point Unit Circle 1, ), ) 10 ), 1 15 1, 0) 180 150 5 0, 1) 1, ) 1, ) 90, ) 0 ) 5, ), 1 0 ), 1 0 5 0 0 0 1, 0) 7 11 5 7 10 5 0 5 ), 1 15 ), 1 0 00, ) 70 1, ) 1, ), ) 0, 1) Unit Circle 0 / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample Find the eact value of each epression. a) sin 7 b) cos c) tan d) sec 70 Unit Circle 1 / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Periodic Functions A number line can be wrapped around a circle infinitel man times, so the domain of both the sine and cosine functions is, ). This means more than one value t will be mapped onto the same point P, ). Graphing ordered pairs of the form t, sin t) shows how the function repeats periodicall. Unit Circle / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Periodic Functions A function = f t) is periodic if there eists a positive real number c such that f t + c) = f t) for all values of t in the domain of f. The smallest number c for which f is periodic is called the period of f. sint + n ) = sin t cost + n ) = cos t tant + n ) = tan t period = period = period = Unit Circle / 5 Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar Eample 7 Use the period of each function to determine an eact value. a) cos 9 ) b) sin c) tan 9 Unit Circle / 5
Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summar What You Learned You can now: Find values of trigonometric functions for an angle. Find the values of trigonometric functions using the unit circle. Do problems Chap. #1, 5, 9-1 odd,, -7 odd, -57 odd, 1-5 odd, 7, 75 Unit Circle 5 / 5