7 4 THE TANGENT FUNCTION Functions Modeling Change
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7. 4 THE TANGENT FUNCTION Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
The Tangent Function Suppose P = (x, y) in the figure is the point on the unit circle specified by the angle θ. We define the function, tangent of θ, or tan θ by tan θ = y / x for x ≠ 0. Since x = cos θ and y = sin θ, we see that tan θ = sin θ/cos θ for cos θ ≠ 0. P = (x, y) 1 θ y x Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
The Tangent Function in Right Triangles If θ is an angle in a right triangle (other than the right angle), c a θ b Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
The Tangent Function in Right Triangles Example 3 The grade of a road is calculated from its vertical rise per 100 feet. For instance, a road that rises 8 ft in every one hundred feet has a grade of Grade = 8 ft/100 ft = 8%. Suppose a road climbs at an angle of 6◦ to the horizontal. What is its grade? Solution From the figure, we see that tan 6◦ = x/100, so, using a calculator, x = 100 tan 6◦ = 10. 510. Thus, the road rises 10. 51 ft every 100 feet, so its grade is 6° 100 ft 10. 51/100 = 10. 51%. x A road rising at an angle of 6◦ (not to scale) Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
Interpreting the Tangent Function as Slope We can think about the tangent function in terms of slope. In the Figure, the line passing from the origin through P has In words, tan θ is the slope of the line passing through the origin and point P. P = (x, y) x (0, 0) y Line has slope y/x = tan θ θ Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
Graphing the Tangent Function • By observation we see y = tan θ has period 180◦. • Since the tangent is not defined when the x-coordinate of P is zero, the graph of the tangent function has a vertical asymptote at θ = − 270◦, − 90◦, 270◦, etc. Domain: All θ ≠ …, − 270◦, − 90◦, 270◦ , … Graph of the tangent function Range: All Reals Θ (degrees) Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally
