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1 §1.6 Trigonometric Review The student will learn about: angles in degree and radian measure, trigonometric functions, graphs of sine and cosine functions, and the four other trigonometric functions.

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2 §1.6 Trigonometric Review What follows is basic information about the trigonometric essentials you will need for calculus. It is not complete and assumes you have a full knowledge of trigonometric functions.

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3 Degrees and Radians Angles are measured in degrees where there are 360º in a circle. Angles are also measured in radians where there are 2π radians in a circle. Example 1. Convert 90º to radians. Degree-Radian Conversion Formula

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4 Degrees and Radians Example 2. Convert π/3 radians to degrees. = 60 º Degree-Radian Conversion Formula Some Important Angles Radian 0π/6π/4π/3π/2π2π Degree 0º0º30º45º60º90º180º360º

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5 Trigonometric Functions Consider a unit circle with center at the origin. Let point P be on the circle and form an angle of θ (in radians) with the positive x axis. θ (1, 0) (0, 1) P (x, y) The sine θ is the ordinate of point p, i.e. y = sine θ. The cosine θ is the abscissa of point p, i.e. x = cosine θ. To find the value of either the sine or cosine functions use the sin and cos keys of your calculator. Make sure you are in the correct mode, [either degrees or radians] pertaining to the problem.

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6 Trigonometric Functions Consider a unit circle with center at the origin. Let point P be on the circle and form an angle of θ (in radians) with the positive x axis. θ (1, 0) (0, 1) P (x, y) The sine θ is the ordinate of point p, i.e. y = sine θ. The cosine θ is the abscissa of point p, i.e. x = cosine θ. Remember the sign “+/-” of the abscissa and the ordinate in the different quadrants. That will help you get their signs correct in the future.

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7 Trigonometric Functions The cosine θ is the abscissa of point p, i.e. x = cosine θ. The sine θ is the ordinate of point p, i.e. y = sine θ. The sine and cosine of some special angles Radian 0π/6π/4π/3π/2π2π Degree 0º0º30º45º60º90º180º360º Sine 01/2 2/2 3/2 100 Cosine 1 3/2 2/2 1/ The values in aqua will repeat. Know them!

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8 Graphs of Sine and Cosine f (x) = sine x is a periodic function that repeats every 2π radians and can be found on your graphing calculator as: f (x) = cosine x is a periodic function that repeats every 2π radians and can be found on your graphing calculator as:

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9 Graphs of Sine and Cosine Being able to picture these graphs in my mind has helped me a lot in determining the numeric value of a trig function. If you combine this information with the basic numerical information given earlier you will be in pretty good shape for getting the correct numerical values for the trig functions. y = sin x y = cos x

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10 Four Other Trig Functions Four Other Trigonometric Functions These functions may also be graphed on your calculator.

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11 Four Other Trig Functions Four Other Trigonometric Functions These functions may also be graphed on your calculator.

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12 Trigonometric Identities There are literally hundreds of trig identities. Several of the most useful follow. Reciprocal identities

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13 Trigonometric Identities Pythagorean Identities sin 2 x + cos 2 x = tan 2 x = sec 2 x 1 + cot 2 x = csc 2 x Quotient Identities

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14 Trigonometric Identities Sum Angle Identities

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15 Trigonometric Identities Double Angle Identities

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16 Trigonometric Identities And lots more Handout

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17 Summary. We defined the six trigonometric functions. We examined the graphs of the trigonometric functions. We are now ready to continue our study of calculus using the trigonometric functions. Degree-Radian Conversion Formula We examined some trig identities.

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18 ASSIGNMENT §1.6; Page 28; 1 – 11.

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