Section 4.6 Graphs of Other Trigonometric Functions

Overview In this section we examine the graphs of the other four trigonometric functions. After looking at the basic, untransformed graphs we will examine transformations of tangent, cotangent, secant, and cosecant. Again, extensive practice at drawing these graphs using graph paper is strongly recommended.

Tangent and Cotangent Three key elements of tangent and cotangent: 1.For which angles are tangent and cotangent equal to 0? These will be x-intercepts for your graph. 2.For which angles are tangent and cotangent undefined? These will be locations for vertical asymptotes. 3.For which angles are tangent and cotangent equal to 1 or -1? These will help to determine the behavior of the graph between the asymptotes.

y = tan x

y = cot x

Transformations |A| = amplitude π/B = period (distance between asymptotes). C/B gives phase shift from zero.

Examples—Graph the Following

Secant and Cosecant The graphs of secant and cosecant are derived from the graphs of cosine and sine, respectively: 1.Where sine and cosine are 0, cosecant and secant are undefined (location of vertical asymptotes). 2.Where sine and cosine are 1, cosecant and secant are also 1. 3.Where sine and cosine are -1, cosecant and secant are also -1.

y = csc x (derived from graph of y = sin x)

y = sec x (derived from graph of y = cos x)

Transformations To graph a transformation of cosecant or secant, graph the transformation of sine or cosine, respectively, then use the reciprocal strategy previously discussed: |A| = amplitude (affects the places where secant or cosecant is equal to 1 or -1) 2π/B = period (distance between asymptotes) C/B = phase (horizontal) shift, left if (+), right if (-)

Examples—Graph the Following