Trigonometric Functions

Transcription

1 Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011

2 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real numbers, evaluate trigonometric functions using the unit circle, use the domain and period to evaluate sine and cosine functions, use a calculator to evaluate trigonometric functions.

3 Unit Circle The unit circle is the graph of the equation x + y = 1. y x, y t t Θ x

4 Arc Length Recall: s = rθ in general. When r = 1 (as in the case of the unit circle) then s = θ. The arc length will be denoted t and is the arc of the circle intercepted by the central angle whose measure is θ. The coordinates of the point (x, y) depend on the value of the arc length t.

5 Trigonometric Functions Definition Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. sin t = y cos t = x tan t = y x, if x 0 cot t = x y, if y 0 sec t = 1 x, if x 0 csc t = 1 y, if y 0

6 Unit Circle Values

7 Table of Values t x y t x y π 1 0 π π π π 1 1 7π 1 π 1 π π 1 π 1 7π 11π 1 1

8 Examples Fill in the missing values in the following table. t cos t sin t tan t cot t sec t csc t π 11π

9 Examples Fill in the missing values in the following table. t cos t sin t tan t cot t sec t csc t 1 π 11π 1

10 Examples Fill in the missing values in the following table. t cos t sin t tan t cot t sec t csc t 1 π 1 11π 1 1

11 Examples Fill in the missing values in the following table. t cos t sin t tan t cot t sec t csc t 1 π π

12 Examples Fill in the missing values in the following table. t cos t sin t tan t cot t sec t csc t 1 π 1 11π

13 Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. The range of the sine and cosine function is the interval [ 1, 1]. Since cos( t) = cos t, cosine is an even function. Since sin( t) = sin t, sine is an odd function.

14 Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. The range of the sine and cosine function is the interval [ 1, 1]. Since cos( t) = cos t, cosine is an even function. Since sin( t) = sin t, sine is an odd function. Question: are the other four trigonometric functions even, odd, or neither?

15 Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. The range of the sine and cosine function is the interval [ 1, 1]. Since cos( t) = cos t, cosine is an even function. Since sin( t) = sin t, sine is an odd function. Question: are the other four trigonometric functions even, odd, or neither? tan( t) = tan t cot( t) = cot t sec( t) = sec t csc( t) = csc t (odd) (odd) (even) (odd)

16 Peiodic Functions Definition A function f is periodic if there exists a positive real number c such that f(t + c) = f(t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

17 Peiodic Functions Definition A function f is periodic if there exists a positive real number c such that f(t + c) = f(t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. cos(t + n π) = cos t sin(t + n π) = sin t for all t when n is an integer. Remark: sine and cosine are periodic functions with period π.

18 Examples Use the period of the trigonometric functions to evaluate the function. cos π = sin 9π = sin 19π = ( cos 9π ) =

19 Examples Use the period of the trigonometric functions to evaluate the function. cos π = 1 sin 9π = sin 19π = ( cos 9π ) =

20 Examples Use the period of the trigonometric functions to evaluate the function. cos π = 1 sin 9π = sin 19π = 1 ( cos 9π ) =

21 Homework Read Section.. Exercises: 1, 5, 9, 1,..., 5, 57