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Graphs of the Sine, Cosine, & Tangent Functions Objectives: 1. Graph the sine, cosine, & tangent functions. 2. State all the values in the domain of a basic trigonometric function that correspond to a given value of the range. 3. Graph the transformations of sine, cosine, & tangent functions. 7.1

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Graphing the Cosine Function on the Coordinate Plane DegreesRadianscos(t) 0°01 30°.866 45°.707 60°.5 90°0 120°-.5 135°-.707 150°-.866 180° 210°-.866 225°-.707 240°-.5 270°0 300°.5 315°.707 330°.866 360°1

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Graphing the Sine Function on the Coordinate Plane

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Characteristics of the Sine & Cosine Functions Period :2π Domain:The set of all real numbers (−∞, ∞) Range:[−1, 1] Function Type: Sine (Odd) Cosine (Even) 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. Remember: Even Functions are symmetric about the y-axis, Odd Functions are symmetric about the origin (shown below).

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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. (1,0) (0,1) (0,-1) (-1,0)0°, 360° 90° 180° 270°

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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. (1,0) (0,1) (0,-1) (-1,0)0°, 360° 90° 180° 270°

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Graphing the Tangent Function on the Coordinate Plane

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

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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.

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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.

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

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

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

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Copyright © Cengage Learning. All rights reserved. 4.5 Graphs of Sine and Cosine Functions.

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