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Unit 3 Part B Trigonometry

Unit 3 Part B Trigonometry. By: Rachel, Jennifer, Navi and Anmol. Key Concepts. In this unit we covered three major concepts which are: - Trig Identities -Graphing y= sin x and y= cos x -Applications of Trigonometry. Trigonometric Identities. Quotient Identity:

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Unit 3 Part B Trigonometry

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  1. Unit 3 Part BTrigonometry By: Rachel, Jennifer, Navi and Anmol

  2. Key Concepts • In this unit we covered three major concepts which are:- Trig Identities-Graphing y=sinx and y=cosx-Applications of Trigonometry

  3. Trigonometric Identities Quotient Identity: The Pythagorean Identity: OR • Meant to prove that the left side and right side of an equation are equal to one another.

  4. Periodic Functions • A function which y-values has a constant and repeating pattern at regular intervals. Properties of a Periodic Function: Cycle:One complete pattern of a periodic function. Period: The time it takes to complete one back-and-forth swing. Amplitude: Half the distance between the maximum and minimum values of a periodic function.

  5. Graphing y=sinx • Find out the period by dividing 360 degree by k value. • To find the amplitude look at your a (vertical stretch) value. • Now that you know the length of your graph and your amplitude, start plotting the three zeroes (origin, half the period value & the period value) • Plot the max value between your first two zeroes (a value) • Plot min value between your last two zeroes, then connect your points.

  6. Graphing y=cosx • Find out the period by dividing 360 degree by k value. • To find the amplitude look at your a (vertical stretch) value. • Now that you know the length of your graph and your amplitude, start plotting your two max values (one at first max value & second at your last max value at the end of your period). • Plot the min value at half of the period. • Plot the two zeroes in between your first max and min and between the last max and min and connect the points into a cosine function.

  7. Vertical Stretches of Periodic Functions • a> 1, graph is expanded vertically by a factor of a • 0 < a < 1, graph is compressed vertically by a factor of a • To graph: Multiply the y-values by a • The amplitude of each function is the numerical value of a

  8. Horizontal Stretches in a Periodic Functions • k> 1, graph is compressed horizontally by a factor of • 0 < k < 1, graph is expanded horizontally by a factor of To graph: Determine the period of the function first by using:

  9. Vertical Translations • Determined by the value of c • If c > 0, graph shifts upc units. • If c < 0, graph shifts downc units.

  10. Horizontal Translations(Phase Shifts) • Determined by the value of d • If d > 0, graph shifts rightd units. • If d < 0, graph shifts leftd units.

  11. Trig Identities Examples • Solve the followingi) tanx + 1 = 1 tanxsinxcosx • ii) cos2x = sin2x + 2cosx-1

  12. Graphing Sine and Cosine Functions • Graph the following:

  13. Application Questions • The ferris wheel at the town fair is 30m tall and makes ones rotation every 40 seconds. The ferris wheel’s loading platform sits at 2m above the ground. • State the period of Ehsan’s ride. • State an equation for a sine function that models Ehsan’s ride on the Ferris wheel.

  14. Continuation From Previous Slide • P = 40 seconds • Equation – Amplitude = 14 K value = 2 Functions = sine • Equation – y= 14 sin (9 (x – 360 ) + 2

  15. Application Question #2 • A sinusoidal function has an amplitude of 5 units, a period of 120 degrees and a maximum at (0,3) • State the equation of the function.

  16. Questionnaire Time! Were you guys actually being attentive? Or you were guys dazing off during our lesson?  Only the questionnaire game will let us experts know the truth!

  17. Question #1 • True or false:Is an identity an equation?

  18. Question #2 • In the function y = sin (x – 45 ) + 2, describe the horizontal translation and the vertical translation.

  19. Question #3 • Before you start graphing, what must you ALWAYS remember to find first and how do you determine that?

  20. Question #4 In the following function: f(x) = 10 sin (x – 45) + 10 Determine the amplitude, the period, the phase shift and the vertical shift of the function!

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