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 = cos θ and y = sin θ, we see that tan θ = sin θ/cos θ for cos θ ≠ 0. 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. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally P = (x, y) θ x y 1

3. The Tangent Function in Right Triangles If θ is an angle in a right triangle (other than the right angle), Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally θ c a b

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%. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Grade = 8 ft/100 ft = 8%. 6°6° x 100 ft A road rising at an angle of 6 ◦ (not to scale)

5. Interpreting the Tangent Function as Slope Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally 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) Line has slope y/x = tan θ (0, 0) x y θ

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. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Graph of the tangent function Θ (degrees) Domain: All θ ≠ …, −270 ◦,−90 ◦, 90 ◦, 270 ◦, … Range: All Reals