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Graphs of Trigonometric Functions and Inverses

Graphs of Trigonometric Functions and Inverses. What you will learn. Evaluate and graph the inverse sine, cosine and tangent functions. Evaluate compositions of trigonometric functions. Plan for the Day. Quick review of graphing Graphing Inverses Evaluating Inverses Compositions

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Graphs of Trigonometric Functions and Inverses

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  1. Graphs of Trigonometric Functions and Inverses

  2. What you will learn • Evaluate and graph the inverse sine, cosine and tangent functions. • Evaluate compositions of trigonometric functions.

  3. Plan for the Day • Quick review of graphing • Graphing Inverses • Evaluating Inverses • Compositions You will need your calculator and unit circle

  4. Quick Review Graphing Review.ppt

  5. Get out your calculator: Try These

  6. Get out your calculator: Try These 22.6o 17.1o 48.2o 50.0o

  7. Inverse Functions

  8. Inverse Functions What is the definition of an inverse function? What kind of test did we use to check for inverse functions? Will the inverses of Sine, Cosine and Tangent be a function?

  9. x 0 cos x 1 0 -1 0 1 y = cos x y x Graph of the Cosine Function Cosine Function To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  10. x x 1 0 0 1 0 -1 cos-1x cos x 1 0 -1 0 1 0 Inverse of the Cosine (cos-1 ) Switch the x and y

  11. y x

  12. Inverse of Cosine (Page 324) On the interval, [0, π] cosine is decreasing On the interval, [0, π] y = cos x takes on its full range of values [-1,1] On the interval, [0, π] cosine is one-to-one So, on this restricted interval, cosine does have an inverse function written as y = arccos x or y = cos-1x “The angle whose cosine is x” The domain of y = arccos x is [-1, 1] and the range is [0, π].

  13. Evaluate the following arccos ½ arccos -1 cos-1 0 arccos (-½)

  14. x 0 sin x 0 1 0 -1 0 y = sin x y x Graph of the Sine Function To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  15. x x 0 0 -1 0 1 0 sin-1x sin x 0 1 0 -1 0 0 Inverse of the Sine (sin-1 ) Switch the x and y

  16. Inverse Sine y x

  17. Inverse of Sine (page 322) On the interval, [-π/2, π/2] sine is increasing On the interval, [-π/2, π/2] y = sin x takes on its full range of values [-1,1] On the interval, [-π/2, π/2] sine is one-to-one So, on this restricted interval, sine does have an inverse function written y = arcsin x or y = sin-1x “The angle whose sine is x” The domain of y = arcsin x is [-1, 1] and the range is [-π/2, π/2].

  18. Evaluate the following arcsin ½ arcsin 1 sin-1 0 arcsin (-½)

  19. To graph y = tan x, use the identity . y Properties of y = tan x 1. domain : all real x x 4. vertical asymptotes: period: Graph of the Tangent Function At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. 2. range: (–, +) 3. period: 

  20. Inverse Tangent y x

  21. Inverse of Tangent (page 324) On the interval, (-π/2, π/2) tangent is increasing On the interval, (-π/2, π/2) y = tan x takes on its full range of values (-∞, ∞) On the interval, (-π/2, π/2) tangent is one-to-one So, on this restricted interval, tangent does have an inverse function written y = arctan x or y = tan-1x “The angle whose tangent is x” The domain of y = arctan x is (-∞, ∞) and the range is (-π/2, π/2).

  22. Evaluate the following arctan 1 arctan -1 tan-1 0

  23. Composite Functions (Page 326) What is a composite function? Remember… f(f-1(x)) = x f-1 (f(x)) = x then… Think about it … what is … arcsin(sin π/3) = ?? cos (arccos ½) = ?? cos (arcsin ½) = ??

  24. Composite Functions (Page 326) If -1≤ x ≤1 and –π/2≤ y ≤ π/2 then sin (arcsin x) = x and arcsin (sin y) = y If -1≤ x ≤1 and 0≤ y ≤ πthen cos (arccos x) = x and arccos (cos y) = y If x is a real number and –π/2< y < π/2 then tan (arctan x) = x and arctan (tan y) = y

  25. Composite Functions (Page 326) Let’s look at this statement If -1≤ x ≤1 and –π/2≤ y ≤ π/2 then sin (arcsin x) = x and arcsin (sin y) = y What if y >π/2 and not within the range stated?? arcsin(sin 2π/3) = ??

  26. Remember completing problems like: Given the cosine of an acute angle is 2/3, find the tangent.

  27. Homework 28 Section 4.7, p. 328: 1-25 every other odd, 37-41 odd, 49, 53

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