Cosine (1.2 continued) Objectives: 1. Determine the range and - PowerPoint PPT Presentation

Domain and Period of Sine and Cosine (1.2 continued) Objectives: 1. Determine the range and period for sine and cosine and use to evaluate problems. 2. Determine and evaluate EVEN and ODD trig functions. 3. Use a calculator to evaluate trig functions.

Domain for all t = all Real Numbers Range: sin t Range: cos t sin t = y cos t = x so, -1< y < 1 so, -1 < x < 1 -1 < sin t < 1 1 < cos t < 1 -1 < sin( t  2  n) < 1 -1 < cos( t  2  n) < 1

Because sin( t  2  n) = sin t and cos( t  2  n) = cos t , sine and cosine are Periodic Functions . Def: f is periodic if there exists c such that f(t+c) = f(t) . ( , ) 01 #periods angle measure sine cosine 1 0  /2  /2 0 1 0  /2+2  1 1 0 2  /2+2(2  ) Co-terminal Angles 1 0  /2+3(2  ) 3

Ex 4: Use the period to evaluate the sine and cosine of each. A. t = 5  Find a co-terminal angle that is on the unit circle.  3     5 2 Not on unit circle, yet.   (  10 sin t  0    , ) 3 2 cos t   1

Ex 4 (cont’d) : Use the period to evaluate the sine and cosine of each. B. t = 8  /3 Find a co-terminal angle that is on the unit circle.  8    − 1 3 8 6  2   2   2 , 2 3 3 3 3 cos t   1 3 sin t  2 2

Ex 4 (cont’d) : Use the period to evaluate the sine and cosine of each. C. t = -15  /2 Find a co-terminal angle that is on the unit circle.   15   11   15 4 Not on unit    2    2 2 circle, yet. 2 2    11 4   7 ( , ) 01 Not on unit   2 2 2 circle, yet.    7 4   3 sin t  1 Not on unit   2 2 2 circle, yet. cos t  0   3 4     2 Finally ! 2 2

Even and Odd Trig Functions Cosine and Secant functions are EVEN. cos(- t ) = cos t sec(- t ) = sec t Sine, Cosecant, Tangent, and Cotangent are ODD. sin(- t ) = -sin( t ) csc(- t ) = -csc( t ) tan(- t ) = -tan( t ) cot(- t ) = -cot( t )

Ex 5: Use the value of the trig function to evaluate the indicated functions. A. sin t = 1/3   1 = − 1   sin t 3 (a) sin(- t ) 3 sin 𝑢 = − 3 1   csc t = − 1 (b) csc(- t )   3

Ex 5(cont’d) : Use the value of the trig function to evaluate the indicated functions. B. cos (- t ) = -1/5 cos t   1   cos( t ) cos t (a) cos t 5   sec( t ) sec t (b) sec(- t ) cos t   5 1   5  1

Ex 6: Use a calculator to evaluate the expression. Round to four decimal places. Be sure calculator is in “radian” mode! A. Sin  /4 B. Cos (-3) 0.7071 -0.9900 1  C. Cos (-1.7) D. Csc 0.8 -0.1288 sin . 0 8 1  13940 .    1.4486 E. Sec 22.8 cos 22 8 . Homework: p. 147-148 #30 - 52 even, except #42