Trigonometry Lecture Notes_part1-1

Even and odd trigonometric functions recall that an

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. Even and Odd Trigonometric Functions Recall that an even function is a function that has the following property: ( ) ( ) f x f x - = Also, recall that an odd function is a function that has the following property: ( ) ( ) f x f x - = - Now, consider two points on the unit circle ( ) : cos ,sin P t t and ( ) : cos( ),sin( ) Q t t - - , I have created a drawing of two such points: Notice how the x value is the same for both points in spite of the fact that the point Q uses the angle – t, but you can also see that the y values have the opposite sign of each other. This diagram helps us to see that cos cos( ) sin( ) sin t t t t = - - = - which means that cosine is an even function, and sine is an odd function.
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Even and Odd Trigonometric Functions The cosine and secant functions are even. cos( ) cos t t - = sec( ) sec t t - = The sine, cosecant, tangent, and cotangent functions are odd. sin( ) sin t t - = - csc( ) csc t t - = - tan( ) tan t t - = - cot( ) cot t t - = - Example 22 Find the exact value of a. cos (- 45 ringoperator ) and b. tan( 3 π - ). Periodic Functions A function f is periodic if there exists a positive number p such that ( ) ( ) f t p f t + = for all t in the domain of f. The smallest number p for which f is periodic is called the period of f. Periodic Properties of the Sine and Cosine Functions ( ) sin 2 sin t t π + = ( ) cos 2 cos t t π + = The sine and cosine functions are periodic functions and have period 2 π . Example 23 Find the exact value of: a. tan 420 ringoperator and b. 9 sin 4 π Periodic Properties of the Tangent and Cotangent Functions ( ) tan tan t t π + = ( ) cot cot t t π + = The tangent and cotangent functions are periodic functions and have period π . Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t,
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( ) sin 2 sin t n t π + = , ( ) cos 2 cos t n t π + = , and ( ) tan tan t n t π + = Section 6.5 Graphs of Sine and Cosine Functions The Graph of sin y x = We can graph the trig functions in the rectangular coordinate system by plotting points whose coordinates satisfy the function. Because the period of the sine function is 2 π , we will graph the function on the interval [0, 2 π ]. Table of values (x, y) on sin y x = X 0 6 π 3 π 2 π 2 3 π 5 6 π π 7 6 π 4 3 π 3 2 π 5 3 π 11 6 π 2 π y 0 1 2 3 2 1 3 2 1 2 0 1 2 - 3 2 - 1 - 3 2 - 1 2 - 0 Below is a graph of the sine curve which uses two periods instead of just the one we have above. From the graph of the sine wave we can see the following things: The domain is the set of all real numbers. The range consists of all numbers from [-1, 1]
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The period is 2 π . This function is an odd function which can be seen from a graph by observing symmetry with respect to the origin. Graphing Variations of y=sinx To graph variations of y = sinx by hand, it is helpful to find x-intercepts, maximum points, and minimum points. There are three intercepts, so we can find these five points by using the following scheme. Let 1 x = the x value where the cycle begins. Then for all i in [2,5] the i th point can be found using: 1 4 i i period x x - = + .
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