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4.2/4.3 Trigonometric Functions: The Unit Circle & Right Triangle Trigonometry Quick Review Give the value of the angle in degrees. 2 1. 3 2. 4 Use special triangles to evaluate. 3. cot 4 7 4. cos 6 5. Use a right triangle to find the other five trigonometric 4 functions of the acute angle given cos 5 Quick Review Solutions Give the value of the angle in degrees. 2 1. 120 3 2. 45 4 Use special triangles to evaluate. 3. cot 1 4 7 4. cos 3/2 6 5. Use a right triangle to find the other five trigonometric 4 functions of the acute angle given cos 5 sec 5 / 4, sin 3 / 5, csc 5 / 3, tan 3 / 4, cot 4 / 3 What you’ll learn about • How to identify a Unit Circle and its relationship to real numbers • How to evaluate Trigonometric Functions using the unit circle • Periodic Functions of sine and cosine functions … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications to model and solve real-life problems. Initial Side, Terminal Side Positive Angle, Negative Angle of a Nonquadrantal Angle θ 1. 2. 3. 4. 5. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. Draw a perpendicular segment from P to the xaxis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. Use the coordinates of point P and the definitions to determine the six trig functions. Trigonometric Functions of any Angle Let be any angle in standard position and let P( x, y ) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P ( x, y ) to the origin, i.e., let r x y . Then 2 y r x cos r y tan ( x 0) x sin r y r sec x x cot y csc 2 ( y 0) ( x 0) ( y 0) Unit Circle The unit circle is a circle of radius 1 centered at the origin. Trigonometric Functions on Unit Circle Let t be any real number, and let P( x, y ) be the point corresponding to t when the number line is wrapped onto the unit circle as described above. Then sin t y cos t x tan t y ( x 0) x 1 y 1 sec t x x cot t y csc t ( y 0) ( x 0) ( y 0) The 16-Point Unit Circle Example Using one Trig Ratio to Find the Others Find sin and cos , given tan 4 / 3 and cos 0. Example Using one Trig Ratio to Find the Others Find sin and cos , given tan 4 / 3 and cos 0. Since tan is positive the terminal side is either in QI or QIII. The added fact that cos is negative means that the terminal side is in QIII. Draw a reference triangle with r 5, x -3, and y -4. sin -4 / 5 and cos -3/ 5 Periodic Function A function y f (t ) is periodic if there is a positive number c such that f (t c) f (t ) for all values of t in the domain of f . The smallest such number c is called the period of the function.