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Trigonometry Lecture Notes_part1-1

This means that the domain for these two functions

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Unformatted text preview: This means that the domain for these two functions must be all real numbers. Now, because the radius of the unit circle is one, we have restrictions on the values that are possible for both y and x, so the ranges will be given by: [ ] 1,1- . That is 1 cos 1 t- ≤ ≤ , and 1 sin 1 t- ≤ ≤ . 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. 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 r ) 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 r 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, ( ) 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....
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