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Warm Up Evaluate. 1 . 2. 3. 4.

6.4 Trigonometric Graphs and Fundamental Identities. Warm Up Evaluate. 1 . 2. 3. 4. 0.5. 0. 0.5. Find the measure of the reference angle for each given angle. 5. 145°. 5. 317°. 35°. 43°. 6.4 Trigonometric Graphs and Fundamental Identities. Objective.

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Warm Up Evaluate. 1 . 2. 3. 4.

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  1. 6.4 Trigonometric Graphs and Fundamental Identities Warm Up Evaluate. 1.2. 3. 4. 0.5 0 0.5 Find the measure of the reference angle for each given angle. 5. 145° 5. 317° 35° 43°

  2. 6.4 Trigonometric Graphs and Fundamental Identities Objective Recognize and graph periodic and trigonometric functions.

  3. 6.4 Trigonometric Graphs and Fundamental Identities Vocabulary periodic function cycle period amplitude frequency phase shift

  4. 6.4 Trigonometric Graphs and Fundamental Identities

  5. 6.4 Trigonometric Graphs and Fundamental Identities

  6. 6.4 Trigonometric Graphs and Fundamental Identities

  7. 6.4 Trigonometric Graphs and Fundamental Identities

  8. Amplitude Phase shift Vertical shift Period 6.4 Trigonometric Graphs and Fundamental Identities You can combine the transformations of trigonometric functions. Use the values of a, b, h, andkto identify the important features of a sine or cosine function. y = asinb(x–h) + k

  9. 6.4 Trigonometric Graphs and Fundamental Identities Periodic functions are functions that repeat exactly in regular intervals called cycles. The length of the cycle is called its period. Examine the graphs of the periodic function and nonperiodic function below. Notice that a cycle may begin at any point on the graph of a function.

  10. 6.4 Trigonometric Graphs and Fundamental Identities

  11. 6.4 Trigonometric Graphs and Fundamental Identities Example Identify whether each function is periodic. If the function is periodic, give the period. The pattern repeats exactly, so the function is periodic. Identify the period by using the start and finish of one cycle. This function is periodic with a period of .

  12. 6.4 Trigonometric Graphs and Fundamental Identities Example Identify whether each function is periodic. If the function is periodic, give the period. a. b. periodic; 3 not periodic

  13. 6.4 Trigonometric Graphs and Fundamental Identities Sine and cosine functions can be used to model real-world phenomena, such as sound waves. Different sounds create different waves. One way to distinguish sounds is to measure frequency.Frequency is the number of cycles in a given unit of time, so it is the reciprocal of the period of a function. Hertz (Hz) is the standard measure of frequency and represents one cycle per second. For example, the sound wave made by a tuning fork for middle A has a frequency of 440 Hz. This means that the wave repeats 440 times in 1 second.

  14. period amplitude Example Use a sine function to graph a sound wave with a period of 0.002 s and an amplitude of 3 cm. Find the frequency in hertz for this sound wave. Use a horizontal scale where one unit represents 0.002 s to complete one full cycle. The maximum and minimum values are given by the amplitude. The frequency of the sound wave is 500 Hz.

  15. period amplitude 6.4 Trigonometric Graphs and Fundamental Identities Example Use a sine function to graph a sound wave with a period of 0.004 s and an amplitude of 3 cm. Find the frequency in hertz for this sound wave. Use a horizontal scale where one unit represents 0.004 s to complete one full cycle. The maximum and minimum values are given by the amplitude. The frequency of the sound wave is 250 Hz.

  16. sin 6.4 Trigonometric Graphs and Fundamental Identities Example Step 5 Graph using all the information about the function. sin x

  17. 6.4 Trigonometric Graphs and Fundamental Identities Example Step 5 Graph using all the information about the function. y cos x  x – cos (x–)

  18. 6.4 Trigonometric Graphs and Fundamental Identities Warm Up Simplify. 1. 2. cos A 1

  19. 6.4 Trigonometric Graphs and Fundamental Identities Objective Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

  20. Substitute cos θ for and sin θ for 6.4 Trigonometric Graphs and Fundamental Identities A derivation for a Pythagorean identity is shown below. x2 + y2 = r2 Pythagorean Theorem Divide both sides by r2. cos2θ + sin2θ = 1

  21. 6.4 Trigonometric Graphs and Fundamental Identities To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps by using the fundamental identities.

  22. 6.4 Trigonometric Graphs and Fundamental Identities Example Prove each trigonometric identity. Choose the right-hand side to modify. Reciprocal identities. Simplify. Ratio identity.

  23. 6.4 Trigonometric Graphs and Fundamental Identities Example Prove each trigonometric identity. Choose the right-hand side to modify. 1 – cot θ = 1 + cot(–θ) Reciprocal identity. Negative-angle identity. = 1 + (–cotθ) Reciprocal identity. = 1 – cotθ Simplify.

  24. 6.4 Trigonometric Graphs and Fundamental Identities Example Prove each trigonometric identity. sin θcot θ = cos θ Choose the left-hand side to modify. cos θ Ratio identity. cos θ = cos θ Simplify.

  25. 6.4 Trigonometric Graphs and Fundamental Identities Example Rewrite each expression in terms of sin θ, and simplify. Pythagorean identity. Factor the difference of two squares. Simplify.

  26. 6.4 Trigonometric Graphs and Fundamental Identities Example Rewrite each expression in terms of sin θ, and simplify. cot2θ csc2θ– 1 Pythagorean identity. Substitute. Simplify.

  27. 6.4 Trigonometric Graphs and Fundamental Identities Lesson Quiz: Part I 1. Using f(x) = cos x as a guide, graph g(x) = 1.5 cos 2x.

  28. 6.4 Trigonometric Graphs and Fundamental Identities Independent Practice Due Tomorrow p.759 #12-17 all p.765 #10-19 all p.775 #8-43 all Turn in p. 775 on Tuesday for a quiz grade. MUST SHOW ALL WORK FOR CREDIT!!!

  29. 6.4 Trigonometric Graphs and Fundamental Identities Lesson Quiz: Part II Prove each trigonometric identity. 1. sinθ secθ = 2. sec2θ = 1 + sin2θ sec2θ = 1 + tan2θ = sec2θ

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