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7.4. THE TANGENT FUNCTION. 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

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## 7.4

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**7.4**THE TANGENT FUNCTION Functions Modeling Change: A Preparation for Calculus, 4th 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 = cosθ and y = sin θ, we see that tan θ = sin θ/cosθ for cosθ≠ 0. P = (x, y) 1 y θ x Functions Modeling Change: A Preparation for Calculus, 4th 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, 4th 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 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 10.51/100 = 10.51%. Grade = 8 ft/100 ft = 8%. x 6° 100 ft A road rising at an angle of 6◦ (not to scale) Functions Modeling Change: A Preparation for Calculus, 4th 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. Line has slope y/x = tan θ P = (x, y) y x (0, 0) θ Functions Modeling Change: A Preparation for Calculus, 4th 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◦, 90◦, 270◦, etc. Domain: All θ ≠ …, −270◦,−90◦, 90◦, 270◦ , … Range: All Reals Graph of the tangent function Θ (degrees) Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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