Section 4.8 Notes

1 st Day

Example 1 Find all unknown values in the figure, where A = 20° and b = 15. (Round to two decimal places.) c b a C A B

c 15 a C A B B = 70° a ≈ 5.46 c ≈ °

Example 2 A ladder 16 feet long leans against the side of a house. Find the height “h” from the top of the ladder to the ground if the angle of elevation of the ladder is 74°. Round to the nearest tenth of a foot.

House 16 ft. 74° h h ≈ 15.4 feet

Example 3 From a point 65 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are 35° and 43°, respectively. Find the height of the steeple. Round to the nearest tenth of a foot.

A B C D m  ABC = 35° m  ABD = 43° h 65 feet

h + AC = AD h = 60.6 – 45.5 h ≈ 15.1 feet

Example 4 From the time a small airplane is 100 feet high and 1600 ground feet from its landing runway, the plane descends in a straight line to the runway. Determine the plane’s angle of descent. Round to the nearest hundredth of a degree.

plane 100 ft ft. x° x ≈ 3.58°

Day 2

The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscillates, rotates, or is moved in a wave motion. Motion of this nature can be described by a sine or a cosine function and is called simple harmonic motion. The number of cycles that occur in one second in a simple harmonic motion problem is called the frequency of the problem.

Example 5 Consider a ball that is bobbing up and down on the end of a spring. Suppose that 10 cm is the maximum distance the ball moves vertically upward and downward from its equilibrium. Suppose that the time it takes for the ball to move from its maximum displacement above zero to its maximum displacement below zero and back is 4 seconds.

period =4 sec.amplitude =10 cm.

Write the equation for this harmonic motion problem in terms of t, time and d, distance.

How many cycles for this problem are done in one second?

Example 6 Find a model for simple harmonic motion that satisfies the specified conditions. Find the frequency of this harmonic motion. Displacement at t = 00 cm Amplitude4 cm Period6 sec

Example 7 Given the equation for simple harmonic motion d = 4cos (6  t). a.Find the maximum displacement. b.Find the frequency. c.Find the value of d when t = 4 d.Find the least positive value of t for which d = 0.

a.maximum displacement = 4 b. c.d = 4cos 24  so d = 4 d.

Trigonometry and Bearings In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fixed north-south line.

N E S W 35° S 35° E N E S W 80° N 80° W N E S W 45° N 45° E

0° N E 90° S 180° 270° W In air navigation, bearings are measured in degrees clockwise from the north. 60° 0° N E 90° S 180° 270° W 225°

Example 8 An airplane flying at 600 miles per hour has a bearing of 152°. After flying for 1.5 hours, how far south and how far east will the plane have traveled from its point of departure? Round to the nearest tenth of a mile.

N E S W 62° 1.5 hr ∙ 600 mph = 900 m s e 900

Example 9 A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port what bearing should be taken? Round to the nearest degree.

N E S W 45 miles 30 miles port b