Hprec6 4

6-4: Trigonometric Functions

© 2007 Roy L. Gover (www.mrgover.com)

Learning Goals:

•Define the Trigonometric functions in terms of the unit circle.

•Define the Trigonometric functions in the coordinate plane.

Important IdeaTrig ratios depend only the angle and not on a point on the terminal side of the angle.

(3,4)

(6,8)

ExampleFind sin ,cos & when the terminal side of the angle passes through (3,4)

tan

(3,4)

(6,8)

Try ThisFind sin ,cos & when the terminal side of the angle passes through (6,8)

tan

(3,4)

(6,8)

Solution

(6,8)2 2 26 8r 10r

8

6

10

8 4sin

10 5

6 3cos

10 5

8 4tan

6 3

Important Idea

( , )x y

r

x

y

opp

cos x

r

hyp

sin y

rhyp

adj

tan oppadj

y

x

See p. 444. of your text

Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12)

Try This

(5,-12)

Solution

5

-121

3

12sin

13

5cos

13

12tan

5

(5,-12)

Important Idea

Trig ratios may be positive or negative

Find sin, cos & tan of the angle whose terminal side passes through the point (-5,-5)

Try This

(-5,-5)

Solution

(-5,-5)

-5

-5

5 2

5 2sin

25 2

5 2cos

25 2

5tan 1

5

Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12)

Try This

(5,-12)

Solution

5

-121

3

12sin

13

5cos

13

12tan

5

(5,-12)

ExampleFind ,sin t cos t& when the terminal side of an angle passes through the given point on the unit circle.

tan t

1 3,

10 10

1

103

101

Important Idea

cosx

tr

siny

tr

tany

tx

In the unit circle, r=1, therefore

1

yy

1

xx

sin t y

cos t xand

Try Thissin tFind ,cos t

when the terminal side of an angle passes through

tan t&

on the unit circle.

3 4,

5 5

Solution4

sin5

t 3cos

5t

335tan

4 45

t

DefinitionCoterminal Angles: Angles that have the same terminal side.

x

y

y

x

Important IdeaTo find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle.

ExampleFind an

angle coterminal with 420°. Findsin 420andcos420

1. Find smallest positive coterminal angle.

3. Apply definition of sin and cos.

Procedure:

2. Draw picture of coterminal angle.

ExampleFind an

angle coterminal

1. Find smallest positive coterminal angle.

3. Apply definition of sin and cos.

Procedure:

2. Draw picture of coterminal angle.

7

4

with

Find the sin and cos.

Important Idea

The trig ratios of a given angle and all its coterminal angles are the same.

Try ThisFind an angle that is coterminal with 780°. Findsin 780 andcos780 .

3sin 780 sin 60

2

1cos780 cos60

2

Try ThisFind an angle that is coterminal with . Find and .

sin( 10 ) sin 0 0 cos( 10 ) cos0 1

10sin( 10 ) cos( 10 )Hint: use the unit circle to find the trig ratio.

Important IdeaIn addition to finding trig ratios of angles ( ), we can also find trig ratios of real numbers in radians (t). Radians may be in terms of

sin4

cos( 2.56) tan3

or just a number, forexample:

Important IdeaThere are times when we must be satisfied with approximate values of trig ratios. At other times, we can find and prefer exact values.

Example

cos( 2.56)Find the approximate value:

Since the degree symbol (°) is not used, this must be radians.

mode

Try ThisUse your calculator in radian mode to approximate the sin, cos and tan. Round to 4 decimal places. Use the signs of the functions to identify the quadrant of the terminal side.

-187

8

2

5

35.6

Definition

sin t is the sin of a number t where t is in radians.

sin t oppositehypotenus

e

y

r

where 2 2r x y

See page 445 of your text.

Definition

cos t is the cos of a number t where t is in radians.

cos t adjacenthypotenus

e

x

r

where 2 2r x y See page 445 of your text.

Definition

tan tis the tan of a number t where t is in radians.

tan t oppositeadjacent

y

x

See page 445 of your text.

Important Idea

cos cos t

The definitions of the trig ratios are the same for angles and radians, for example:sin sin t

hypopp y

r

hypadj x

r

ExampleFind

the exact value:

cos45 45

cos4

10

10

4

10

10

ExampleFind

the exact value:

sin30 1

sin6

3

16

3

30°

Definition

Reference Angle: the angle between a given angle and the nearest x axis. (Note: x axis; not y axis). Reference angles are always positive.

Important IdeaHow you find the reference angle depends on which quadrant contains the given angle.

The ability to quickly and accurately find a reference angle is going to be important in future lessons.

ExampleFind the reference angle if the given angle is 20°.

In quad. 1, the given angle & the ref. angle are the same.

x

y

20°

ExampleFind the reference angle

if the given angle is .

x

y 9

9

In quad. 1, the given angle & the ref. angle are the same.

ExampleFind the reference angle if the given angle is 120°.For given

angles in quad. 2, the ref. angle is 180° less the given angle.

?120°x

y

ExampleFind the reference

angle if the given

angle is .

?x

y

2

3

2

3

For given angles in quad. 2, the ref. angle is less the given angle.

ExampleFind the reference angle if the given angle is .

x

y

7

6

7

6

For given angles in quad. 3, the ref. angle is the given angle less

Try ThisFind the reference angle if the given angle is

7

4

For given angles in quad. 4, the ref. angle is less the given angle.

2

7

4

4

Try ThisFind the reference angle if the given angle is

x

y 4

Hint: Don’t forget the definition.

4

Important IdeaThe trig ratio of a given angle is the same as the trig ratio of its reference angle except, possibly, for the sign.

ExampleFind the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator.

135°

Procedure1.Sketch the given angle.2.Find and sketch the reference angle. Label the sides using special angle facts.

3.Find sin, cos and tan using definition.

4.Add the correct sign.

ExampleFind the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator.

7

6

Try ThisFind the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator.

60

2

Solution60

3

1

3sin 60

2

1cos60

2

tan 60 3

Important Ideax or y can be positive or negative depending on the quadrant but the hypotenuse ( r ) is always positive.


11

6

Solution 11

6

-13

2

11 1sin

6 2

11 3

cos6 2

11 1 3tan

6 33


4

3

Solution 4

3

-1

23

4 3sin

3 2

4 1cos

3 2

4tan 3

3

The unit circle is a circle with radius of 1. We use the unit circle to find trig functions of quadrantal angles.

-1 1

-1

1

1

Definition

The unit circle

-1 1

-1

1

1

Definition

(1,0)

(0,1)

(-1,0)

(0,-1)

x y

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles:

The x values are the terminal sides for the cos function.

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles:

The y values are the terminal sides for the sin function.

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles :

The tan function is the y divided by the x

-1 1

-1

1

Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot

0°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot90

°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1

Find the values of the six trig functions of the given angle in standard position.

2

Exampl

e

sincostan

cscseccot

-1 1

-1

1


2

Example

sincostan

cscseccot

-1 1

-1

1


3Try This

sincostan

cscseccot

-1 1

-1


Example

sincostan

cscseccot540°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1


Example

sincostan

cscseccot270°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1


7

2

Try This

sincostan

cscseccot

-1 1

-1


Try This

sincostan

cscseccot360°

(1,0)

(0,1)

(-1,0)

(0,-1)

Important Ideas•Trig functions of quadrantal angles have exact values.

•Trig functions of all other angles have approximate values.

•Trig functions of special angles have exact values.

Example

Use a calculator to approximate cos 710° to 4 decimal places.

Don’t forget to check “Mode”.

Example

Use a calculator to approximate sin(72°30’30”) to 4 decimal places.

Example

Use a calculator to approximate csc 15° to 4 decimal places.

Lesson Close

How do you evaluate the trig ratios of quadrantal angles?

Hprec6 4

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