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Graphs of the Sine, Cosine, & Tangent Functions

Graphs of the Sine, Cosine, & Tangent Functions. 7.1. Objectives: Graph the sine, cosine, & tangent functions. State all the values in the domain of a basic trigonometric function that correspond to a given value of the range. Graph the transformations of sine, cosine, & tangent functions.

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Graphs of the Sine, Cosine, & Tangent Functions

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  1. Graphs of the Sine, Cosine, & Tangent Functions 7.1 Objectives: Graph the sine, cosine, & tangent functions. State all the values in the domain of a basic trigonometric function that correspond to a given value of the range. Graph the transformations of sine, cosine, & tangent functions.

  2. Graphing the Cosine Function on the Coordinate Plane

  3. Graphing the Sine Functionon the Coordinate Plane

  4. Characteristics of the Sine & Cosine Functions Period : 2π Domain: The set of all real numbers (−∞, ∞) Range: [−1, 1] Function Type: Sine (Odd) Cosine (Even) Remember:Even Functions are symmetric about the y-axis, Odd Functions are symmetric about the origin (shown below). The period of a function is the amount of time or length of a complete cycle. In other words, how long until the graph starts repeating. For the sine and cosine functions, the period is the same.

  5. Example #1 • State all values of t for which sin(t) = 1. Remember that sine, the y-coordinate, is 1 at 90°. Any angle coterminal with that is also a solution. (0,1) 90° (-1,0) 180° 0°, 360° (1,0) 270° (0,-1)

  6. Example #2 • State all the values of t for which cos(t) = Remember that cosine, the x-coordinate, is -½ at 120° and 240°. Any angle coterminal with those are also a solutions. (0,1) 90° (-1,0) 180° 0°, 360° (1,0) 270° (0,-1)

  7. Graphing the Tangent Functionon the Coordinate Plane

  8. Characteristics of the Tangent Function Period: π Domain: All real numbers except odd multiples of Range: All real numbers (−∞, ∞) Function Type: Odd

  9. Example #3 • State all values of t for which tan(t) = 1. Tangent is 1 where sine and cosine values are the same. This occurs at 45° and 225°. The cycle is shorter for tangent though, so to specify all solutions we only need to add 180° to our original solution.

  10. Basic Transformations of Sine, Cosine, & Tangent • Vertical Stretches Vertical stretches or compressions by a factor of “a”. • Reflections Reflections over the x-axis. • Vertical Shifts Vertical shifting of “b” units.

  11. Example #4 • List the transformation(s) and sketch the graph. Vertical stretch by a factor of 2.

  12. Example #5 • List the transformation(s) and sketch the graph. Vertical compression by a factor of 1/3 and x-axis reflection.

  13. Example #6 • List the transformation(s) and sketch the graph. Vertical shift of four units down.

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