Lesson 5. Section 2.2: Trigonometric Functions of an Acute Angle 1 = 1

Transcription

1 Lesson 5 Diana Pell March 6, 2014 Section 2.2: Trigonometric Functions of an Acute Angle 1 = We can divide 1 into 60 equal parts, where each part is called 1 minute, denoted 1 (so that 1 minute is 1 60 of a degree). One second, 1 is 1 60 of a minute. There are 60 seconds in every minute. 1 = 60 1 = 60 Exercise 1. Change to degrees and minutes. Exercise 2. Change to decimal degrees. 1

2 Section 2.4: Applications Exercise 3. The two equal sides of an isosceles triangle are each 24 cm. If each of the two equal angles measure 52, find the length of the base and the altitude jpg 2.bb Definition 1. An angle measured from the horizontal up is called an angle of elevation. An angle measured from the horizontal down is called an angle of depression. 2

3 Exercise 4. If a 75.0-foot flagpole casts a shadow 43.0 feet long, what is the angle of elevation of the sun from the tip of the shadow? jpg 2.bb 3

4 Exercise 5. A man climbs 213 meters up the side of a pyramid and finds that the angle of depression to his starting points is How high off the ground is he? jpg 2.bb Definition 2. The bearing of a line l is the acute angle formed by the north-south line and the line l. The notation used to designate the bearing of a line begins with N or S (for north or south), followed by the number of degrees in the angle, and ends with E or W (for east or west). 4

5 Exercise 6. San Luis Obispo, CA, is 12 miles due north of Grover Beach. If Arroyo Grande is 4.6 miles due east of Grover Beach, what is the bearing of San Luis Obispo from Arroyo Grande? jpg 2.bb 5

6 Exercise 7. A boat travels on a course of bearing N E for a distance of 238 miles. How many miles north and how many miles east has the boat traveled? jpg 2.bb 6

7 Exercise 8. Figure below is a diagram that shows how Diane estimates the height of a flag pole. She can t measure the distance between herself and the flagpole directly because there is a fence in the way. So she stands at point A facing the pole and finds the angle of elevation from point A to the top of the pole to be Then she turns 90 and walks 25.0 feet to point B, where she measures the angle between her path and a line from B to the base of the pole. She finds that angle is Use this information to find the height of the pole jpg 2.bb 7

8 Section 2.5: Vectors: A Geometric Apporach Note: Quantities that have magnitude and direction are called vector quantities, while quantities with magnitude only are called scalars. Note: Two vector are equivalent if they have the same magnitude and direction. Addition of Vectors Subtraction of Vectors 8

9 Exercise 9. A boat is crossing a river that runs due north. The boat is pointed due east and is moving through the water at 12 miles per hour. If the current of the rivers is a constant 5.1 miles per hour, find the actual course of the boat through the water to two significant digits. Horizontal and Vertical Vector Components Standard Positon for a Vector Horizontal vector component: V x Vertical vector component: V y 9

10 Definition 3. If V is a vector in standard position and θ is the angle measured from the positive x-axis to V, then the horizontal and vertical vector components of V are given by V x = V cos θ and V y = V sin θ Exercise 10. The human cannonball is shot from a cannon with an initial velocity of 53 mi/hr at an angle of 60 from the horizontal. Find the magnitudes of the horizontal and vertical components of the velocity vector. Exercise 11. An arrow is shot into the air so that its horizontal velocity is 25 feet per second and its vertical velocity is 15 feet per second. Find the velocity of the arrow. 10

11 Exercise 12. A boat travels 72 miles on a course of bearing N 27 E and then changes its course to travel 37 miles at N 55 E. How far north and how far east has the boat traveled on this 109-mile trip? Force Definition 4. When an object is stationary (at rest), we say it is in a state of static equilibrium. When an object is in this state, the sum of the forces acting on the object must be equal to the zero vector 0. 11

12 Exercise 13. Danny is 5 years old and weighs 42.0 pounds. He is sitting on a swing when his sister Stacey pulls him and the swing back horizontally through an angle of 30.0 and then stops. Find the tension in the ropes of the swing and the magnitude of the force exerted by Stacey jpg 2.bb 12

13 Work Definition 5. When a constant force F is applied to an object and moves the object in a straight line a distance d, then the work W performed by the force is W = ( F cos θ) d where θ is the angle between the force F and the line of motion of the object. Exercise 14. A shipping clerk pushed a heavy package across the floor. He applies a force of 64 pounds in a downward direction, making an angle of 35 with the horizontal. If the package is moved 25 feet, how much work is done by the clerk? jpg 2.bb 13

14 Homework (due March 11): 2.4: 5, 9, 11, 13, 17, 19, 21, 23, 27, 29, 33, , 19, 23, 25, 29, 31, 33, 35, 39, 43, 47 Chapter 2 Test: 1, 3, 6-8, 11, 12, 17 14