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Trigonometric Functions: Unit Circle and Properties

Learn about trigonometric functions defined on the unit circle, such as sine, cosine, and tangent. Explore domain, range, even and odd functions, reciprocal and Pythagorean identities, periodic properties, and repetitive behavior.

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Trigonometric Functions: Unit Circle and Properties

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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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