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).

/2 #periods angle measure sine cosine /2 1 /2+2  2 /2+2(2) 3 /2+3(2)

( , ) 01

1 1 1 1

Co-terminal Angles

Ex 4: Use the period to evaluate the sine and cosine of each.

• A. t = 5

5 2   

Find a co-terminal angle that is on the unit circle.

 3

Not on unit circle, yet.

3 2      ( , ) 10 sint  0 cost  1

• B. t = 8/3

Ex 4 (cont’d): Use the period to evaluate the sine and cosine of each.

8 3 2   

Find a co-terminal angle that is on the unit circle.

  8 3 6 3    2 3 

sint  3 2 cost   1 2

− 1 2 , 3 2

• C. t = -15/2

Ex 4 (cont’d): Use the period to evaluate the sine and cosine of each. Find a co-terminal angle that is on the unit circle.

  15 2 2  

   15 2 4 2  

 11 2 

Not on unit circle, yet.

  11 2 4 2  

  7 2 

Not on unit circle, yet.

  7 2 4 2     3 2 

Not on unit circle, yet.

  3 2 4 2     2 Finally!

( , ) 01

sint 1 cost  0

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

(a) sin(- t) (b) csc(- t)

 sint

  1 3

 csct

 3

= − 1 3 = − 1 sin 𝑢 = − 3 1

• B. cos (- t) = -1/5

Ex 5(cont’d): Use the value of the trig function to evaluate the indicated functions.

(a) cos t (b) sec(- t)

cos( ) cos   t t

cost   1 5

sec( ) sec   t t

 1 cost   5 1

 5

Ex 6: Use a calculator to evaluate the

• expression. Round to four decimal places.
• A. Sin /4
• B. Cos (-3)
• C. Cos (-1.7)
• D. Csc 0.8
• E. Sec 22.8

Homework: p. 147-148 #30 - 52 even, except #42

Be sure calculator is in “radian” mode! 0.7071

• 0.9900
• 0.1288

 1 0 8 sin .

13940 .

 1 22 8 cos .

 1.4486