Section 6.1

Trigonometric Functions of Acute Angles

Consider a right triangle.

The “right” angle takes up 90 degrees

A triangle has 180 degrees, les the 90 for the right angle.

This leaves 90 degrees for the two remaining angles

So, the 2 remaining angles must be acute angles, ie <90 degrees each.

The ratios of the lengths of these “sides” define the 6 trigonometric ratios.

The side opposite the right angle is called the hypotenuse

The side touching theta is called the adjacent side

The side opposite angle thetaIs called the opposite side

Notice: If we would have used the OTHER acute angle, the names for the“opposite” and “adjacent” sides would have changed.

Let theta be an acute angle of a right triangle. Thenthe six trigonometric functions of theta are:

sine hypotenuseopposite

sin

cosine hypotenuseadjacent

cos

tangentadjacentopposite

tan

cosecantoppositehypotenuse

csc

secantadjacenthypotenuse

sec

cotangentoppositeadjacent

cot

There is some “intelligence” in the particular arrangement of these functions.

For now, notice that reciprocals are horizontally opposite each other.

SOHCAHTOA AOHAHO

817

15

______sin

_____cos

_____tan

_____csc

_____sec

_____cot

Find the Trig functions on theta

RecIprocals

csc1sin

sec1cos

cot1tan

sin1csc

cos1sec

tan1cot

Reciprocal Functions

Similar triangles have the same 3 angles and their sides are proportional.

The Trigonometric functions are based on the angles

1 23

5

4

915

12

53sin 1

54cos 1

43tan 1

53

159sin 2

54

1512cos 2

43

129tan 2

So, the 6 trigfunctions on the same angles willbe the same iftheir sides areproportional.

What do we do if we know the valueof one trig function and are told to find

the other 5?

2524sin

hypotenuseopposite

sinWe know

Label the parts we know on the triangle.

2425

We also know the Pythagorean Equation.

222 cba

Where a and b are sides and c is the hypotenuse.

Find the unknown part

222 2524 b

625576 2 b

492 b

7bThe last side is 7

Show the other 5 trig functions.

What do we do if we know the valueof one trig function and are told to find

the other 5?

2tan

2

1

Think of this as the rational number12tan

222 cba

222 21 c

25 c

5c

5

2tan

552

52sin

55

51cos

21cot

5sec

25csc

Note: Rationalized denominators

Special Triangles

4545

Isosceles Triangle

Relative ratio of sides

1 1

2

Knowing this, we can find exact values for the six trig functions of 45

22

2145sin

22

2145cos

11145tan

145cot

245sec

245csc

KNOW THESEEXACT VALUES

Special Triangles

2

2

2

Equilateral Triangle

60 60

60 If we drop a perpendicular bisectorfrom the top angle we get

30

60901

2

222 cba

222 21 b

32 b

3b

30

60901

23

We now know the particulars of a 30-60-90 triangle

As a result we can now do exact values for 30 and 60 degree trig functions.

2130sin

2330cos

33

3130tan 330cot

332

3230sec

230csc

2360sin

2130cos

330tan

23

3130cot

230sec

332

3230csc

KNOW THESEEXACT VALUES

Angle measures

Currently we are using degrees There are two ways to express degrees

Decimal expression 8268.47 Degrees, minutes, and seconds "36'4947

1 degree= 60 minutes1 minute= 60 seconds

ConversionsTake the decimal part and multiply

by 60 minutes/degree

'608.491'608268.

Take the decimal part and multiplyby 60 seconds/minute

"48.36'1"60'608.

"36'4947

Take the seconds part and divide by60 to convert it to minutes

'6.4947"6036'4947

Take the minutes part and divide by60 to convert it to degrees

8267.47'606.4947

Note: Due to rounding there is a slight difference

It is best to use whatever unit is given TI-84

TI-84 Mode setting to degrees

8268.47 2nd angle

4

2245sin

Exact Answer

7071067812.45sin

Approximate Answer

Complementary Angles – 2 angels whose sum is 90 degrees.

Supplementary Angles – 2 angels whose sum is 180 degrees.

Cofunction Identities

90

Moe

Sam

Jan

Assign some arbitrary names

JanMoe

HO

sin

JanSam

HA

cos

SamMoe

AO

tan

MoeSam

OA

cot

SamJan

AH

sec

MoeJan

OH

csc

JanSam

HO

)90sin(

JanMoe

HA

)90cos(

MoeSam

AO

)90tan(

SamMoe

OA

)90cot(

MoeJan

AH

)90sec(

SamJan

OH

)90csc(

The cofunction ID’s are Complementary 90)90(

)90cos(sin

)90sin(cos

)90cot(tan

)90sec(csc

)90csc(sec

)90tan(cot

Cofunction Identities

cot____52tan csc____13sec

Homework

• 3,5,7,9,13,17,21,23,31,33,41,47,83,85,95