Graphs of Other Trigonometric Functions 11-2 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

Warm Up If sin A = , evaluate. 1. cos A 2. tan A 3. cot A 4. sec A 5. csc A

Objective Recognize and graph trigonometric functions.

The tangent and cotangent functions can be graphed on the coordinate plane. The tangent function is undefined when θ = + n, where n is an integer. The cotangent function is undefined when θ = n. These values are excluded from the domain and are represented by vertical asymptotes on the graph. Because tangent and cotangent have no maximum or minimum values, amplitude is undefined. To graph tangent and cotangent, let the variable x represent the angle θ in standard position.

Like sine and cosine, you can transform the tangent function.

Example 1: Transforming Tangent Functions Using f(x) = tan x as a guide, graph Identify the period, x-intercepts, and asymptotes. g(x) = Step 1 Identify the period. Because b = the period is Step 2 Identify the x-intercepts. The first x-intercept occurs at x = 0. Because the period is 3, the x-intercepts occurs at 3n where n is an integer.

Example 1 Continued Step 3 Identify the asymptotes. Because b = , the asymptotes occur at Step 4 Graph using all of the information about the function.

Check It Out! Example 1 Using f(x) = tan x as a guide, graph . Identify the period, x-intercepts, and asymptotes. Step 1 Identify the period. Because b = the period is Step 2 Identify the x-intercepts. The first x-intercept occurs at x = 0. Because the period is 2, the x-intercepts occur at 2n where n is an integer.

Check It Out! Example 1 Continued Step 3 Identify the asymptotes. Step 4 Graph using all of the information about the function.

Example 2: Graphing the Cotangent Function Using f(x) = cot x as a guide, graph . Identify the period, x-intercepts, and asymptotes. Step 1 Identify the period. Because b = 3 the period is Step 2 Identify the x-intercepts. The first x-intercept occurs at x = . Because the period is , the x-intercepts occurs at , where n is an integer.

Example 2: Graphing the Cotangent Function Step 3 Identify the asymptotes. Because b = 3, the asymptotes occur at Step 4 Graph using all of the information about the function.

Check It Out! Example 2 Using f(x) = cot x as a guide, graph g(x) = –cot2x. Identify the period, x-intercepts, and asymptotes. Step 1 Identify the period. Because b = 2 the period is . Step 2 Identify the x-intercepts. The first x-intercept occurs at x = . Because the period is , the x-intercepts occurs at , where n is an integer.

Check It Out! Example 2 Continued Step 3 Identify the asymptotes. Because b = 2, the asymptotes occur at x = Step 4 Graph using all of the information about the function.

Recall that sec θ = . So, secant is undefined where cosine equals zero and the graph will have vertical asymptotes at those locations. Secant will also have the same period as cosine. Sine and cosecant have a similar relationship. Because secant and cosecant have no absolute maxima, no minima, amplitude is undefined.

You can graph transformations of secant and cosecant by using what you learned in Lesson 14-1 about transformations of graphs of cosine and sine.

Example 3: Graphing Secant and Cosecant Functions Using f(x) = cos x = as a guide, graph g(x) = Identify the period and asymptotes. Step 1 Identify the period. Because sec is the reciprocal of cos the graphs will have the same period. Because b = for cos the period is

Example 3 Continued Step 2 Identify the asymptotes. Because the period is 4, the asymptotes occur at where n is an integer. Step 3 Graph using all of the information about the function.

Check It Out! Example 3 Using f(x) = sin x as a guide, graph g(x) = 2csc x. Identify the period and asymptotes. Step 1 Identify the period. Because csc x is the reciprocal of sin x the graphs will have the same period. Because b = 1 for csc x the period is

Check It Out! Example 3 Continued Step 2 Identify the asymptotes. Because the period is 2, the asymptotes occur at Step 3 Graph using all of the information about the function.

Lesson Quiz: Part I 1. Using f(x) = tan x as a guide, graph g(x) = Identify the period, x-intercepts, and asymptotes. period: 2; x-intercepts: 2n; asymptotes: x =  + 2n

Lesson Quiz: Part II 2. Using f(x) = sin(x) as a guide, graph g(x) = Identify the period, and asymptotes. period: 6; asymptotes: x = 3n