Section 7-4 Evaluating and Graphing Sine and Cosine

Objectives: To use the reference angles, calculators and tables and special angles

to find the values to find the values of Sine and Cosine and graph their functions

Reference AnglesObjective: Find the reference angle of a rotation and use it to find

trigonometric function values DefinitionThe reference angle for a rotation is the acute angle formed by the terminal side and the x-axis

Ѳ = 115o

Reference angle =180-115 = 65o

TerminalSide Ѳ = 225o

Reference angle =225-180 = 45o

Example 12

30o 30o

150o

22

1 1

3 3

Reference Angle

Find the sine , cosine , and tangent of 1320o

We can subtract multiples of 360o We do this by 1320 by 360 and taking the larger part . Thus we subtract 3 multiples of 360.1320 – 3(360) = 240 and 240 – 180 =60

180o 60o

1320o or 240o

o

o

o

3sin 1320

21

cos1320 2

tan 1320 = 3

Example 2

Express 695o in terms of a reference angle 695o - 360o = 335o The reference angle for 335o I s 360o - 335o = 25o

And Since 695o is in the fourth quarter then Sin 695o = - Sin 25o

(0,r)

(-r,0)

(0,-r)

(r,0)

90o

180o

270o

Terminal Side on an AxisIf the terminal side of an angle falls on an axis

0 0

0 0

0 0

0sin 0 =0 sin 90 =1

0cos 0 1 cos90 0

0tan 0 = 0 tan 90 = ( )

0

y y r

r r r rx r x

r r r ry

ux r

x r yndefined

Using Calculators or Tables

The easiest way to find the sine or cosine of most angles is to use a scientific calculator. Always be sure to check whether the calc is in degree or radian mode. If you do not have a calculator you can use the table at the back of this book on page 800

Find the value of each expression to four decimal places.

a. Sin 122o b. Cos 237o c. Cos 5o d. Sin(-2)

Special AnglesObjective: Find the length of sides in special triangles

In a 45-45-90 right triangle the legs are the same length . Lets consider such a triangle whose legs have length 1. Then its hypotenuse has length c

1

1

2

When we split a square In half diagonally we create a 45-45-90 Right triangle.

o

o

o

1 2sin 45 =

22

1 2cos 45 =

221

tan 45 = 11

A 30-60-90 Right TriangleWhen we split an equilateral triangle

in half we create 2 30-60-90 right triangles.

22

1 1

30o

60o

30o

12

3

o

o

o

1sin 30 =

2

3cos 30 =

2

1 3tan 30 =

33

o

o

o

3sin 60 =

21

cos 60 = 2

tan 60 = 3

Example : In this triangle find sinѲ, cos Ѳ, and tan Ѳ

5

3

4

Ѳ

side opposite 3sin = =

5

side adjacent to 4cos =

5

opposite 3tan = =

side adjacent to 4

hypotenuse

hypotenuse

side

Example 2: In ΔABC, b =40cm and <A = 60o. What is the length of side c.

60

c

b

o

bcos A =

40 =cos 60 substitu g tin

c

c

o1 1using cos 60

40 ==

2

2 coc = 80

B

AC

Graph of the Sine Curve

The Cosine Function

Graphs of Sine and CosineTo Graph the Sine Function plot the position of the function using the radian or degree value.

0o

90o 180o 360o

1

-1

Period and Amplitude of Trig Functions

• Amplitude is range of Maximum and Minimum points on a graph

So for function y = A SinBx Amplitude = |A| period

2

B

Try These

= 3 Sin 2xy

Amplitude = |A| = |3| = 3

Period 2 2 =

2B

A = 3 and B= 2

1.) y= Cos 4x

12.) y = 4sin x

23.) y = 6 Cos 4x

14.) y= 3 sin

25.) y = -3sin 2 x

x

The Unit Circle

Homework (1-17) odd Pg.279Day 2: 19,24,27,29,31,33 Pg.280

Graphs of the Six Trig Functions