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Welcome to the Unit 1 seminar of Intro to Linear Algebra! My name is Cathy Johnson. I am here to help you the next 10 weeks! I am looking forward to this term and hoping you are too. I’d like to begin by reviewing a few items from the syllabus, going over ground rules for seminar, and seeing if you have any immediate questions for me. Everyone: Questions before we begin? Syllabus: please read the syllabus which is the same whether you downloaded it from Doc Sharing, read it under Course Home: Syllabus, or read the one I emailed you on the first day of class. Key things include attendance requirements, due dates and late policies, showing your work in your Project assignments, and doing your own work. There have been problems with plagiarism in recent quarters, so all instructors are on the lookout, and the consequences are harsh. How do you contact me? My AIM id is cjohnsonkap. If I am logged into AIM, you are always welcome to send me questions and I will answer if possible. Whenever you AIM me, make sure to identify yourself. The username "bearsfan123" doesn't tell me who you are. If there is a specific time you would like me to meet you online please email me and I will do my best to help you. Email address: My Kaplan email address is ”[email protected]". Make sure my address is added to your address book so that your spam blocker doesn't accidentally block my emails to you. Seminar: Held Wednesdays 8-9 PM ET Due dates and late assignments: All projects, discussion question posts, and seminar option II, are due at 11:59 pm ET the Tuesday that concludes each Unit. All Unit 1 assignments are due this Tuesday. Late projects and seminar option II assignments will be accepted, but penalized 2% per day late. Late message board posts will be scored down 10% per week late. Late projects and discussion board posts will not be accepted after 11:59 pm ET Sunday of unit 10. Extraordinary Circumstances: If you have extraordinary circumstances that prevent you from participating in class, you need to contact me. Good communication is the key to overcoming obstacles you may encounter during the term. I am more than willing to work with you, but I have to know about the situation to do that. I also want to mention that I will post my notes and PowerPoint to the doc sharing folder each week. In addition, you can print the “history” of the seminar so there is no need to take complete notes. Everyone: Questions? Ok, let’s get started on the material. As we discuss the material, please feel free to let me know if you have any questions. This first unit deals with • angles and their measure, • conversion of angle measure from radians to/from degrees, • trigonometric functions, and • right triangle trigonometry. Basic Definitions • A ray is a half-line that begins at a certain point and extends forever in one direction. • The point at which the ray begins is called the endpoint. • When a ray is rotated about its fixed end point, it moves from it initial position to it final position. As it does this it creates an angle between the two positions. • Angles can be measured in degrees. A complete revolution of the ray from its initial position back to its initial position is 360 degrees. • A degree can be subdivided and can be written as either a decimal or a fraction. For example, ½ degree or 0.5 degree. Another system of subdivision is called the sexagesimal system. In that system 1 degree is equal to 60 minutes and 1 minute is equal to 60 seconds. Symbolically, this is written as 1° = 60’ and 1’ = 60” To convert between the a decimal (or fractional) part of a degree to this system, you would use the following conversions. 1° = 60’ = 3600” Everyone: Questions? Converting Angle Decimal and Sexagesimal Measures To convert, use the following: 1° = 60’ = 3600” Example: Solution: Convert 28.6 degrees to the sexagesimal system. 28.6 degrees = 28 ° 0.6(60)’ = 28 ° 36’ Everyone: Questions? Everyone: Convert 123.4 degrees to the sexagesimal system. Converting Angle Decimal and Sexagesimal Measures To convert, use the following: 1° = 60’ = 3600” Example: Solution: Convert 28.6 degrees to the sexagesimal system. 28.6 degrees = 28 ° 0.6(60)’ = 28 ° 36’ Everyone: Questions? Everyone: Convert 123.4 degrees to the sexagesimal system. Solution: 123.4 degrees = 123 ° 0.4(60)’ = 123 ° 24’ Example: Convert 42 ° 36’ 41” to degrees 42 ° 36’ 41” = (42 + 36/60 + 41/3600) ° = 42.6114 ° Everyone: Questions? Everyone: Convert 21 ° 14’ 20” to degrees Example: Convert 42 ° 36’ 41” to degrees 42 ° 36’ 41” = (42 + 36/60 + 41/3600) ° = 42.6114 ° Everyone: Questions? Everyone: Convert 21 ° 14’ 20” to degrees 21 ° 14’ 20” = (21 + 14/60 + 20/3600) ° = 21.2389 ° Everyone: Questions? One other unit for measuring angles is called the radian. If you let the fixed point of the array be the center of a circle, then the angle that is created when the arc length is equal to the radius length is equal to 1 radian. The angle created in one full revolution of the ray is equal to 2π radians. 1 revolution = 2π radians = 360 ° 1 rad same length Converting Degree and Radian Angle Measures To convert between degrees and radians set up a proportion using the following: 1 revolution = 2π radians = 360 ° Example: Solution: Convert 30 degrees to radians 30 degrees = x radians 360 degrees 2π radians Solve: x radians = (30 degrees/360 degrees) 2π radians = (1/12) 2π radians = π / 6 radians Everyone: Questions? Everyone: Convert 75 degrees to radians Everyone: Convert 75 degrees to radians Solution: 75 degrees = x radians 360 degrees 2π radians Solve: x radians = (75 degrees/360 degrees) 2π radians = (5/24) 2π radians = 10π / 6 radians Everyone: Questions? Everyone: Convert 1.47 radians to degrees Everyone: Convert 1.47 radians to degrees Solution: x degrees 1.47 radians = 2 π radians x degrees 360 degrees = ( 1.47 radians / 2 π radians) 360 degrees = 84.2 degrees Everyone: Questions? Trigonometric Functions r a y x sine α = sin α = y/r = opposite side / hypotenuse cosine α = cos α = x/r = adjacent / hypotenuse tangent α = tan α = y/x = opposite / adjacent cotangent α = cot α = x/y = adjacent / opposite secant α = sec α = r/x = hypotenuse / adjacent cosecant α = csc α = r/y = hypotenuse / opposite Pythagorean Theorem r2 = x2 + y2 α r y x Example: Find the values of the 6 trig functions for α. α r 7.2 3.4 First, determine r. r2 = (7.2)2 + (3.4)2 = 51.84 + 11.56 = 63.4 r = 7.96 Example: Determine the value of the six trigonometric functions for the angle α. r = 7.96 a sin α cos α tan α cot α sec α csc α y = 7.2 x = 3.4 = = = = = = y/r x/r y/x x/y r/x r/y = = = = = = 7.2 / 7.96 3.4 / 7.96 7.2 / 3.4 3.4 / 7.2 7.96 / 3.4 7.96 / 7.2 Everyone: Questions? = = = = = = 0.905 0.427 2.118 0.472 2.341 1.106 Everyone: Determine the value of the six trigonometric functions for the angle α. r a 6 4 Everyone: Determine the value of the six trigonometric functions for the angle α. r a 6 4 r = sqrt( 62 + 42 sin α = y/r = cos α = x/r = tan α = y/x = cot α = x/y = sec α = r/x = csc α = r/y = ) = 7.21 6 / 7.21 4 / 7.21 6/4 4/6 7.21 / 4 7.21 / 6 Everyone: Questions? = = = = = = 0.832 0.555 1.5 0.667 1.803 1.202 Evaluating Trigonometric Functions Okay, let’s use a calculator to evaluate trigonometric functions when the angle is known and measured in both degrees and radians. make sure your calculator is on degree mode cos (23 °) = 0.921 make sure your calculator is on radian mode cos (4.17 rad) = -0.516 Everyone: Questions? The procedure for the sine and tangent functions would be similar. However, the remaining three trigonometric functions are not as direct. For those, you need the following reciprocal identities. cot α = 1 / tan α tan α = 1 / cot α sec α = 1 / cos α cos α = 1 / sec α csc α = 1 / sin α sin a = 1 / csc α Let’s try a few examples. sec (23 °) = 1 / cos (23 °) = 1 / 0.921 = 1.086 sec (4.17 rad) = 1 /(-0.516) = -1.937 You can use the basic trig definitions and the Pythagorean theorem to solve right triangles. Let’s continue with an earlier problem and show how to identify all sides and angles of the following right triangle. r = 7.96 y = 7.2 α x = 3.4 To find the value of the angle α, you’ll use the inverse cosine function on your calculator. It is indicate as follows: cos-1x Everyone: Questions? r = 7.96 7.2 α 3.4 Now, let’s use the value for cos α that we found earlier, 0.427. cos-1(0.427) = 1.13 rad = 64.72 ° Since the sum of the interior angles of a triangle is 180 degrees, the remaining angle will be 90° – 64.72° = 25.28 ° Everyone: Questions? Everyone: Solve the following right triangle. (all sides, all angles) r β α 4.8 5.1 Everyone: Solve the following right triangle. (all sides, all angles) r β 5.1 α 4.8 r = sqrt(5.12 + 4.82) = 7.004 cos α = 4.8/7.004 = 0.6853 cos-1 (0.6853) = 46.74 ° Β = (90 – 46.74) ° = 43.26 ° Everyone: Questions? Applications Right triangles are often used to find heights and distances of objects. To do so, sketch a right triangle, label the parts of the triangle that are given, and solve for the remaining parts that are unknown. Definition: Angle of Elevation The angle that the line of sight from the top of an object makes with the horizontal is called its elevation. Definition: Angle of Depression When an object from a lower level is viewed from a vantage point, the angles that the line of sight makes with the horizontal is called the angle of depression. Please look at Example 17 in the handout given in the course shell. Everyone: Questions?