0.2: Trignometric FunctionsObjective:•Understand

relationships between angles and sides of right triangles•Review trig functions of special & quadrantal angles

©2008 Roy L. Gover (www.mrgover.com)

Calculus Prerequisite Knowledge

DefinitionThe special angles are:

•30°,60°, and 45° in degrees

• and in radians , ,

6 3 4

In calculus, we use radians.

Important Idea

These angles are special because they have exact value trig functions.

Consider the first two special angles…

Long Side

Short

Sid

e

Hypotenus

e 6

3

Lon

g Sid

e

Hypotenuse 6

3

Short Side

…orientation does not change the relationships between sides and angles

Lon

g Sid

eHyp

oten

use 6

3

Short Side

…orientation does not change the relationships between sides and angles

Analysis

Consider an equilateral triangle with sides 2 units long

2

2

2

Analysis...equilater

al triangles are equiangular and each angle must be

2

2

2

3

3

3

3

Analysis

…the angle bisector intersects the base at the midpoint

2 2

1 13

3

6

6

Analysis

2 2

1 1

1

2?

?= 3

6

6

6

3

3

3

Try ThisFind the height of an equilateral triangle with each side 4 inches long.

4 4

4

?

Important IdeaIn a rad triangle:, ,

6 3 2

•the short side is one-half the hypotenuse.

•the long side is times the short side.

3Memoriz

e

Try ThisFind the length of the missing sides:

6

10

5

5 3

Try This

Find the length of the missing sides:

3

5

5 3

3

10 3

3

Try ThisWhat is wrong with this picture? 6

Consider the last special angle…

6

3

4

4

4

Hypot

enus

e

…orientation does not change the relationships between sides and angles

4

4

Hypotenuse

…orientation does not change the relationships between sides and angles

4

4

…and sides opposite equal angles are equal... 4

4

x

x

and by the2x

pythagorean theorem, the hypotenuse is...

Important IdeaIn a triangle:

•The legs of the triangle are equal.

•the hypotenuse is times the length of the leg.

Memoriz

e2

, ,4 4 2

Try ThisFind the length of the missing sides

4

4

2

22 2

Try ThisFind the length of the missing sides

2

2

24

4

Review

•

-1 1

-1

1

x

yr

oppositesin

hypotenuse

y

r

adjacentcos

hypotenuse

x

r

oppositetan

adjacent

y

x

•

• Memorize

Review

•

-1 1

-1

1

x

yr

hypotenusecsc

opposite

r

y

hypotenusesec

adjacent

r

x

adjacentcot

opposite

x

y

•

• Memorize

Find the exact values of the six trig

3

-1 1

-1

1

Example

functions of the given special angle in standard position.

-1 1

-1

1

Analysis

1. Make a right triangle.

rads3

-1 1

-1

1

Analysis

2. Let the short side be 1, then use your knowledge of special angles

32

1

to find the measures of each side of the right triangle.

rads3

-1 1

-1

1

Analysis

32

1

3. Use the definitions of the trig functions to find each trig function exact value.

rads3

-1 1

-1

1

Analysis

32

1

opp 3sin

hyp 2

adj 1cos

hyp 2

opp 3tan 3

adj 1

rads3

-1 1

-1

1

Analysis

32

1

hyp 2 3csc

opp 3

hypsec 2

adj

adj 3cot

opp 3

rads3

Find the exact values of the six trig

-1 1

-1

1Try This

functions of the given special angle in standard position.

4

Solution2

sin4 2

2cos

4 2

tan 14

csc 24

sec 24

cot 14

Important IdeaIf the given angle is other than a quadrant I angle, the exact values of the trig functions are calculated using the reference angle of the given angle.

Review

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

Review

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

Review

2

1

-1

-2

-2 -1 1 2 3

ReferenceAngle

GivenAngle &

Review2

1

-1

-2

-3 -2 -1 1 2

GivenAngleReference

Angle

Review

-3 -2 -1 1 2

2

1

-1

-2

ReferenceAngle

GivenAngle

Review

-3 -2 -1 1 2

2

1

-1

-2

ReferenceAngle

GivenAngle

ExampleDraw a diagram showing the relevant angles and sides and find the sin, cos & tan of

5

6

Given Angle

Ref. Angle6

5

6

Example

5

6

6

Short Side

Long Side

Given Angle

Example

5

6

6

2

1

3

Why is 1 & 2 pos. & neg.?

3

5 1sin

6 2

5 3cos

6 2

5 3tan

6 3

Try ThisDraw a diagram showing the relevant angles and sides and find the csc, sec & cot of

5

6

Given Angle

Ref. Angle6

5

6

Solution

5

6

6

2

1

3 5csc 2

6

5 2 3sec

6 3

5cot 3

6

-1 1

-1

1Find the values of the six trig functions of the given angle in standard position.

2

sincostan

cscseccot

Quadrantal Angles

-1 1

-1

1Find the values of the six trig functions of the given angle in standard position.

2

sincostan

cscseccot

Quadrantal Angles

-1 1

-1

1Find the values of the six trig functions of the given angle in standard position.

sincostan

cscseccot

Quadrantal Angles

-1 1

-1

1Find the values of the six trig functions of the given angle in standard position.

3

2

sincostan

cscseccot

Quadrantal Angles

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 to 4 decimal places.

You did not forget to check “mode”, did you???

4

9

Example

Use a calculator to approximate csc(11.55) to 4 decimal places.

Check “mode” every time.

Lesson Close

You will frequently use the trig knowledge you learned in Precalculus in Calculus

Assignment

Worksheet 0-2