Pages From Algebra And Trigonometry 9th Edition-2.pdf 4n5m2j

1. Calculating the Time of a Trip From a parking lot you want to walk to a house on the ocean. The house is located 1500 feet down a paved path that parallels the beach, which is 500 feet wide. Along the path you can walk 300 feet per minute, but in the sand on the beach you can only walk 100 feet per minute. See the illustration. Ocean Beach

!

500 ft x Forest

Paved path 1500 ft Parking lot

(a) Calculate the time T if you walk 1500 feet along the paved path and then 500 feet in the sand to the house. (b) Calculate the time T if you walk in the sand directly toward the ocean for 500 feet and then turn left and walk along the beach for 1500 feet to the house. (c) Express the time T to get from the parking lot to the beachhouse as a function of the angle u shown in the illustration. (d) Calculate the time T if you walk directly from the parking lot to the house. 500 c Hint: tan u = d 1500 (e) Calculate the time T if you walk 1000 feet along the paved path and then walk directly to the house. (f) Graph T = T1u2. For what angle u is T least? What is x for this angle? What is the minimum time? 1 (g) Explain why tan u = gives the smallest angle u that is 3 possible. 2. Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. (a) Express the length L 3 ft of the line segment shown as a function of the angle u. (b) Discuss why the length of the longest ladder ! that can be carried L around the corner is 4 ft equal to the smallest value of L. 3. Electrical Engineering A resistor and an inductor connected in a series network impede the flow of an alternating current. This impedance Z is determined by the reactance X of the inductor and the resistance R of the resistor. The three quantities, all measured in ohms, can be represented by the sides of a right triangle as illustrated, so Z2 = X2 + R2. The angle f is called the phase angle. Suppose a series network has an inductive reactance of X = 400 ohms and a resistance of R = 600 ohms. (a) Find the impedance Z. (b) Find the values of the six trigonometric functions of the phase angle f.

Z

X

" R

4. Electrical Engineering Refer to Problem 3. A series network has a resistance of R = 588 ohms. The phase 5 angle f is such that tan f = . 12 (a) Determine the inductive reactance X and the impedance Z. (b) Determine the values of the remaining five trigonometric functions of the phase angle f. 5. Geometry Suppose that the angle u is a central angle of a circle of radius 1 (see the figure). Show that u (a) Angle OAC = 2 (b) ƒ CD ƒ = sin u and ƒ OD ƒ = cos u u sin u (c) tan = 2 1 + cos u C ! A

O

D B

6. Geometry Show that the area A of an isosceles triangle is A = a2 sin u cos u, where a is the length of one of the two equal sides and u is the measure of one of the two equal angles (see the figure). a

a

!

!

7. Geometry Let n Ú 1 be any real number and let u be any p angle for which 0 6 nu 6 . Then we can draw a triangle 2 with the angles u and nu and included side of length 1 (do you see why?) and place it on the unit circle as illustrated. Now, drop the perpendicular from C to D = 1x, 02 and show that x =

tan1nu2

tan u + tan1nu2 y

C O

h n!

! D = (x, 0) (1, 0)

x

8. Geometry Refer to the figure. The smaller circle, whose radius is a, is! tangent to the larger circle, whose radius is b. The! ray OA contains a diameter of each circle, and the ray OB is tangent to each circle. Show that cos u =

10. Geometry Refer to the figure, where a unit circle is drawn. The line segment DB is tangent to the circle. y 1

2ab a + b 2

C

(This shows that cos u equals the ratio of the geometric mean of a and b to the arithmetic mean of a and b.) [Hint: First show that sin u = 1b - a2>1b + a2.]

O

B

(a) Express the area of ¢OBC in of sin u and cos u.

a

A

O

9. Geometry Refer to the figure. If ƒ OA ƒ = 1, show that 1 (a) Area ¢OAC = sin a cos a 2 1 (b) Area ¢OCB = ƒ OB ƒ 2 sin b cos b 2 1 (c) Area ¢OAB = ƒ OB ƒ sin1a + b2 2 cos a cos b (e) sin1a + b2 = sin a cos b + cos a sin b (d) ƒ OB ƒ =

[Hint: Area ¢OAB = Area ¢OAC + Area ¢OCB] B C # " O

D 1

x

–1

b !

B 1

! –1

D

A

[Hint: Use the altitude from C to the base OB = 1.] (b) Express the area of ¢OBD in of sin u and cos u. 1 (c) The area of the sector OBC of the circle is u, where u 2 is measured in radians. Use the results of parts (a) and (b) and the fact that Area ¢OBC 6 Area of sector OBC 6 Area ¢OBD to show that 1 6

1 u 6 sin u cos u

11. If cos a = tan b and cos b = tan a, where a and b are acute angles, show that sin a = sin b =

B

3 - 25 2

12. If u is an acute angle and tan u = x, express the remaining five trigonometric functions in of x.