Aim: How can we graph the reciprocal trig functions using the three basic trig ones? Do Now: In the diagram below of right triangle JMT, JT = 12, JM = 6 and mJMT = 90. What is the value of cot J? J M T

Reciprocal Identities Co- Co- Co-

Trig Values in Coordinate Plane y Quadrant II Quadrant I function reciprocal function reciprocal cos  is – sin  is + tan  is – sec  is – csc  is + cot  is – cos  is + sin  is + tan  is + sec  is + csc  is + cot  is + x Quadrant III Quadrant IV cos  is – sin  is – tan  is + sec  is – csc  is – cot  is + cos  is + sin  is – tan  is – sec  is + csc  is – cot  is – For any given angle, a trig function and its reciprocal have values with the same sign.

Reciprocals – Graph of Cosecant reciprocal of 0 - undefined therefore these are the only points of equality f(x) = csc x

Reciprocals – Graph of Secant reciprocal of 0 undefined therefore these are the only points of equality f(x) = sec x

Reciprocals – Graph of Cotangent the only points of equality f(x) = cot x -

Model Problems Which expression represents the exact value of csc 60o? Which expression gives the correct values of csc 60o? Which is NOT an element of the domain of y = cot x?

Model Problems A handler of a parade balloon holds a line of length y. The length is modeled by the function y = d sec , where d is the distance from the handler of the balloon to the point on the ground just below the balloon, and  is the angle formed by the line and the ground. Graph the function with d = 6 and find the length of the line needed to form an angle of 60o.

Model Problem Graph the function |a| = amplitude (vertical stretch or shrink) h = phase shift, or horizontal shift k = vertical shift |b| = frequency dilation frequency phase shift vertical shift a = 2 b = 3 k = -2

Model Problem Graph the function dilation frequency phase shift vertical shift a = 2 b = 3 k = -2