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Trigonometric Functions: The Unit Circle 4.2

Trigonometric Functions: The Unit Circle 4.2. Unit Circle. The unit circle is a circle of radius 1 with its center at the origin. If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t , then.

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Trigonometric Functions: The Unit Circle 4.2

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  1. Trigonometric Functions: The Unit Circle4.2

  2. Unit Circle • The unit circle is a circle of radius 1 with its center at the origin.

  3. If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then Definitions of the Trigonometric Functions in Terms of a Unit Circle

  4. Points on the Unit Circle

  5. Text Example Use the Figure to find the values of the trigonometric functions at t=/2. Solution: The point P on the unit circle that Corresponds to t= /2 has coordinates (0,1). We use x=0 and y=1 to find the Values of the trigonometric functions

  6. The Domain and Range of the Sine and Cosine Functions • The domain of the sine function and the cosine function is the set of all real numbers • The range of these functions is the set of all real numbers from -1 to 1, inclusive.

  7. Trigonometric Functions at /4.

  8. The cosine and secant functions are even. cos(-t) = cos t sec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(-t) = -sin t csc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t Even and Odd Trigonometric Functions

  9. Example • Use the value of the trigonometric function at t = /4 to find sin (- /4 ). Solution: sin(-t) = -sin t, so sin(- /4 ) = -sin(/4 ) = -2/2

  10. Reciprocal Identities

  11. Quotient Identities

  12. Example • If sin t = 2/5 and cos t = 21/5, find the remaining four trig functions

  13. Pythagorean Identities

  14. Example • Find cos t if sin t = 4/5

  15. 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. Definition of a Periodic Function

  16. sin(t + 2) = sin t and cos(t + 2) = cos t The sine and cosine functions are periodic functions and have period 2. Periodic Properties of the Sine and Cosine Functions sin  = sin 3

  17. tan(t + ) = tan t and cot(t + ) = cot t The tangent and cotangent functions are periodic functions and have period . Periodic Properties of the Tangent and Cotangent Functions tan  = tan 2

  18. Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t, sin(t + 2n) = sin t, cos(t + 2n) = cos t, and tan(t + n) = tan t

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