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14-2

Graphs of Other Trigonometric Functions. 14-2. Warm Up. Lesson Presentation. Lesson Quiz. 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.

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14-2

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  1. Graphs of Other Trigonometric Functions 14-2 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  2. Warm Up If sin A = , evaluate. 1.cos A2. tan A 3. cot A 4. sec A 5. csc A

  3. Objective Recognize and graph trigonometric functions.

  4. 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.

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

  6. Using f(x) = tan x as a guide, graph Identify the period, g(x) = x-intercepts,and asymptotes. Because b = the period is Example 1: Transforming Tangent Functions Step 1 Identify the period. 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.

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

  8. Because b = the period is 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. 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.

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

  10. Step 1 Identify the period. Because b = 3 the period is 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 Using f(x) = cot x as a guide, graph . Identify the period, x-intercepts, and asymptotes. Step 2 Identify the x-intercepts.

  11. 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.

  12. 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 Using f(x) = cot x as a guide, graph g(x) = –cot2x. Identify the period, x-intercepts, and asymptotes. Step 1 Identify the period.

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

  14. 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.

  15. 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.

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

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

  18. 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 Using f(x) = sin x as a guide, graph g(x) = 2csc x. Identify the period and asymptotes.

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

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

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

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