Chapter 6 Applications of Trigonometry

6.1 VECTORS IN THE pLANE

Quick Review

Quick Review

Quick Review Solutions

Quick Review Solutions

Directed Line Segment

Two-Dimensional Vector

Initial Point, Terminal Point, Equivalent

Find the vector for both RS= <3,4> QP= <3,4>

Magnitude

Example Finding Magnitude of a Vector

Example Finding Magnitude of a Vector

Find Vector Find the magnitude of v represented by 𝑆𝑇 , where S=(2, -8) and T= (-3, 7)

Vector Addition and Scalar Multiplication

Example Performing Vector Operations

Example Performing Vector Operations

Group Work Let u=<-1,3> and v=<5, -6> Find A) u+v B) 3u C) 2u+(-1)v

Unit Vectors

Example Finding a Unit Vector

Example Finding a Unit Vector

Find the unit vector P=<3,9> Q=<1, 6>

Standard Unit Vectors

Resolving the Vector

Example Finding the Components of a Vector

Example Finding the Components of a Vector

Example Finding the Direction Angle of a Vector

Example Finding the Direction Angle of a Vector

Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is speed.

Word Problem The pilot pilots the plane from San Franciso due east. There is a 65 mph wind with the bearing 60 degrees (from the y-axis). Find the compass heading the pilot should follow, and determine what the airplane’s ground speed will be (assuming its speed with no wind is 450 mph).

Answer Bearing should be approx 94.14 degrees 𝜃=−4.14° Speed is 505.12 mph

Word Problem A jet is flying on a bearing of 65° at 500 mph. Find the component form of the velocity of the airplane. Recall that the bearing is the angle that the line of travel makes with due north, measured clockwise.

Answer <453.15,211.31>

Homework Practice Pg 511 #1-45 eoo

Polar Coordinates

Quick Review

Quick Review Use the Law of Cosines to find the measure of the third side of the given triangle. 4. 40º 8 10 5. 35º 6 11

Quick Review Solutions

Quick Review Solutions Use the Law of Cosines to find the measure of the third side of the given triangle. 4. 40º 8 10 5. 35º 6 11 6.4 7

The Polar Coordinate System

Example Plotting Points in the Polar Coordinate System

Example Plotting Points in the Polar Coordinate System

Finding all Polar Coordinates of a Point

Coordinate Conversion Equations

Example Converting from Polar to Rectangular Coordinates

Example Converting from Polar to Rectangular Coordinates

Example Converting from Rectangular to Polar Coordinates

Example Converting from Rectangular to Polar Coordinates

Example Converting from Polar Form to Rectangular Form

Example Converting from Polar Form to Rectangular Form

Example Converting from Polar Form to Rectangular Form

Example Converting from Polar Form to Rectangular Form

6.4 Polar Coordinates Page 537 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

6.4 Polar Coordinates (cont’d) Page 537 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Homework Practice Pg 539 #1-50 eoe

Limits and Motion: The tangent problem

Quick Review

Quick Review Solutions

What is Tangent?

Average Velocity Average velocity is the change in position divided by the change in time.

Limits at a (Informal)

Example Finding the Slope of a Tangent Line

Example Finding the Slope of a Tangent Line

Example: A ball rolls down a ramp so that its distance s from the top of the ramp after t seconds is exactly feet. What is its instantaneous velocity after 3 second?

Average Rate of Change

Derivative at a Point

Derivative at a Point (easier for computing)

Example Finding a Derivative at a Point

Example Finding a Derivative at a Point

Derivative

Example Finding the Derivative of a Function

Example Finding the Derivative of a Function

Example: Find if

Example: Find if

Homework Practice P 801 #1-32 eoe

Integral: The area problem

Quick Review

Quick Review Solutions

Example Computing Distance Traveled A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?

Example Computing Distance Traveled A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?

Limits at Infinity (Informal)

Definite Integral