77. Geometry A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is 16 feet from the streetlight and 5 feet from the tip of the streetlight's shadow, the person's shadow starts to appear beyond the streetlight's shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the streetlight. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the streetlight? 78. Geometry A 25-meter line is used to tether a helium- filled balloon. Because of a breeze, the line makes an angle of approximately 750 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? 79. Why you should learn it (p. 273) A zip-line steel cable is being constructed for a reality television competition show. The high end of the zip-line is attached to the top of a 50-foot pole while the lower end is anchored at ground level to a stake 50 feet from the base of the pole. 50 ft 1 50 ft -(a) Find the angle of elevation of the zip-line. (b) Find the number of feet of steel cable needed for the zip-line. (c) A contestant takes 6 seconds to reach the ground from the top of the zip-line. At what rate is the contestant moving down the line? At what rate is the contestant dropping vertically? 80. Geomatics In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.50. After you drive 13 miles closer to the mountain, the angle of elevation is 9 (see figure). Approximate the height of the mountain. -7 3.50 90 13 mi Not drawn to scaleProcedures and Problem Solving Evaluating Trigonometric Functions In Exercises 7-10, Evaluating Trigonometric Functions In Exercises find the exact values of the six trigonometric functions of 13-20, sketch a right triangle corresponding to the the angle 0 shown in the figure. (Use the Pythagorean trigonometric function of the acute angle 0. Use the Theorem to find the third side of the triangle.) Pythagorean Theorem to determine the third side of the triangle and then find the other five trigonometric 8. 13 functions of 0. 5 14. cot 0 = 5 16. cos 0 = 10. 18. csc 0 = 17 4 18 20. sin 0 = 12 Evaluating Trigonometric Functions In Exercises 11 and 12, find the exact values of the six trigonometric functions of the angle 0 for each of the triangles. Explain why the function values are the same. 12. 15 7.5 4