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# 7.4

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

## 7.4

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1. 7.4 THE TANGENT FUNCTION Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

2. 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

3. 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

4. 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

5. 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

6. 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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