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4.01 Angles and Their Measure Keywords Angle Definitions/Examples Initial side: Terminal Side: Vertex: Copy 3 images from lesson Standard Position: Positive Angle: Coterminal Angles: Negative Angle: Radian Measures We measure an angle by stating the amount of rotation from the initial side to the terminal side. Radian: Acute Angles: Obtuse Angles: Degree Measure Convert between Degrees and Radians To convert from degrees to radians: To convert from radians to degrees: Test your skills Application: Find arc length Formula for arclength: If the radius of a circle is 4", what is the length of the arc measured by Pi symbol/2 radians? 4.02 Trigonometric Functions of Acute Angles Keywords Right Triangle Definitions/examples Hypotenuse: Opposite side: Adjacent Side: Six Trigonometric Functions http://youtu.be/5tp74g4N8EY Test your skills Determine the six trigonometric functions for the following right triangles. Choose the Check Your Answers link below to view the solutions to these problems. adjacent side = 6 and opposite side = 8 adjacent side = 4 and opposite side = 4 hypotenuse = 2 and adjacent side = 1 Special Trigonometric Values you should Memorize Trigonometric Identities Reciprocal Identities: Quotient Identities Pythagorean Identities Test your skills Answer each of the following. Choose the Check Your Answers link below to view the solutions to these problems. a. In a right angle, an acute has cos = .6. Determine sin and tan. b. A visitor to the Washington Monument is curious about the height of the monument. He walks away from the monument until his line of sight to the top of the monument is exactly 60°. He has taken 107 steps to this point and each step is 3 feet long. How high in feet is the Washington Monument? 4.03 Trigonometric Functions: The Unit Circle Keywords The Unit Circle Definitions/Examples Definition of Trigonometric Functions Let x be a real number and (x,y) the point on the unit circle corresponding to t. Sin(t)= Cos(t)= Tan (t)= Csc (t)= Sec(t)= Test your Skills Cot(t)= Determine the values of the six trigonometric functions for the following. a. t = 4π /3 b. t=3π /4 Domain and Period of Sine and Cosine Definition of a Periodic Function Test your skills Determine the values of the six trigonometric functions for the following. a. t=15π/6 b. t= 11π/2 4.04 Trigonometric Functions of Any Angle Keywords Definitions of Trigonometric Functions of any Angle Definitions/examples Let be an angle in standard position with (x,y) a point on the terminal side of and r= Sin= Cos= Tan= Cot= Sec= Csc= Example: If an angle has a terminating side which contains point (1, √3), what are the values of the sine, cosine, and tangent of ? Definition/Examples: Trigonometric Functions of Real Numbers Test your skills: Answer each of the following. A. For an angle with the point (-3, -4) on its terminating side, what are the values of the sine, cosine, and tangent? B. For an angle with the point (-1,-1) on its terminating side, what are the values of the sine, cosine, and tangent? C. If tan = -1, and the sintheta > 0, what is the value of cos? D. Determine the values of cos Pi, sin(3Pi/2), and tan(2Pi). *** Check your answers within the lesson ***** Definition: Reference Angles If the angle is in the 2nd quadrant, then the reference angle is _______________ For an angle in the 3rd quadrant, the reference angle is _________________ And for an angle _______________ in the 4th quadrant, the reference angle is Test Your Skills Determine the reference angle for the following. a. 100° b. 225° c. -100° *** Check your answers within the lesson ***** Definitions/Examples: Trigonometric Functions of Real Numbers ASTC A S T C 4.05a Graphs of Sine and Cosine Functions Keywords: Definitions/Examples Basic Sine and Cosine Curves Sine Curve: Key points on Maximum and Minimum points: a Sine or Cosine Curve X-axis intercepts Y-axis intercept The Standard Form of the Equations for Sine and Cosine Sine and Cosine curves can be expressed in the following standard equations: Amplitude Period Phase Shift Vertical Shift Summary Test your skills: Determine the amplitude, period, and left and right endpoints for the following. ***Check your answers within the lesson***** Graphing by Hand How to graph by hand: Test your skills Graph each of the following trigonometric curves by hand. Find the key points to help you plot an accurate curve. You can check your answers by graphing the functions with your calculator (be sure that your calculator is in radian mode). Use the trace key to check the values of the intercepts and the maximum and minimum values. Test your skills ***Check your answers within the book*** 4.06 Graphs of other Trigonometric Functions Keywords The Standard Form of the Tangent Function Definitions/ examples Definitions/ examples/notes About the Graph of the Tangent Function Definitions/ examples/notes Summary of How to Graph the Tangent by Hand 1. 2. 3. 4. 5. Graph y = 2 tan (3x + pi) 1. 2. 3. 4. 5. Test your skills For each of the following tangent curves, list the new period, the phase shift, equation of the vertical asymptotes, and the intercepts. Graph of the Cotangent Function Definitions/ examples/notes Graphs of the Reciprocal Functions Definitions/ examples/notes Graphing Review Definitions/ examples/notes 4.07 Inverse Trigonometric Functions Keywords Definitions/Examples/Notes Inverse Sine Function Definitions/Examples/Notes: For this domain, the following properties exist: 1. 2. 3. Definition of Inverse Sine Function The inverse sine function is defined by Test your skills: Evaluate each of the following without using a calculator. ***Check your answers within the lesson*** Other Inverse Trigonometric Functions Definitions of the Inverse Trigonometric Functions Domain Range Test your skills Evaluate each of the following without using a calculator. Remember to pay attention to the range of the inverse function, and give your angle in the correct quadrant. a. arccos(1) b. cos-1(0) c. arccos(0) d. arctan(-1) e. tan-1(√3) **Check your answers within the lesson*** Compositions of Functions Definitions/Examples/Notes: Example: Test your skills Find the value of each of the following. You can check your answers by typing the expression into your calculator. 1. sin ( sin-1 (1)) 2. cos (sin -1 (-0.5)) 3. tan-1 (sin /2) 4.08 Solving Problems with Trigonometry Keywords Definitions/Examples/Notes Applications Involving Triangles Definitions/Examples/Notes : Simple harmonic motion Definitions/Examples/Notes: Harmonic Motion A point that moves on a coordinate line is said to be in simple harmonic motion if its distance d from the origin at time t is given by either