aim: the six trigonometric functions course: alg. 2 & trig. aim: what does sohcahtoa have to do...

Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.

Aim: What does SOHCAHTOA have to do with our study of right triangles?

Do Now:

Key terms: adjacent, opposite & hypotenuse

4

3

5

C B

AWhat are the following ratios?

ACAB =

ACCB =

BCAB =

3545

43


Trigonometry Basics - Sine

In a right ABC with right angle BCA

• The sine of angle B, written sine B, is defined as

hypotenusetheoflength

Boppositelegtheoflength

BA

ACB sin

4

3

5

C B

Asin B =

ACAB

=54

3sin

5

BCA

AB


Sine’s Reciprocal

What is the reciprocal of sin?What is the reciprocal of 3? 1/3

1/sin

the reciprocal of sin has a special name:

sin =

2

2csc = ?

2

2=

2 2

2

2ex.

using the calculator to find csc 53º:find csc 53º:

sin x -153 ENTER ENTER

Display: 1.252135658

Method 1

cosecant

NOTE: csc sin 1


Trigonometry Basics - Cosecant


• The cosecant of angle B, written csc B, is defined as

cscBA length of the hypotenuse

BAC length of the leg opposite B

4

3

5

C B

Acsc B =

ABAC

=45

5csc

3

ABA

BC


Trigonometry Basics - Cosine


• The cosine of angle B, written cos B, is defined as

hypotenusetheoflength

Btoadjacentlegtheoflength

BA

BCB cos

4

3

5

C B

Acos B =

BCAB

=53

4cos

5

ABA

AC

Recall: 3

sin5

BCA

AB

the sine of an acute angle has the same value as the cosine of its complement.


Cosine’s Reciprocal

The reciprocal of cosine is the secant :

sec 1

cos

sec = ? ex. cos =

1

22

= ? 2nd ÷cos -1 ENTER1 2

Display: 60

using the calculator to find sec :Find sec (-38º):

( – )1 cos ENTER

Display: 1.269018215

Method 2

÷ 38

NOTE: sec cos 1


Trigonometry Basics - Secant


• The secant of angle B, written sec B, is defined as

4

3

5

C B

Asec B =

ABBC

=35

sec

AB length of the hypotenuseB

BC length of the leg adjacent B

5sec

4

ABA

AC


Trigonometry Basics - Tangent

• The tangent of angle B, written tan B, is defined as

Btoadjacentlegtheoflength

Boppositelegtheoflength

BC

ACB tan

4

3

5

C B

A


tan B =ACBC

=34

3tan

4

BCA

AC


Tangent’s Reciprocal

The reciprocal of tan is the cotangent :

cot 1

tan

cot = ? ex. tan =

3

3

3

3

3=

= ? 2nd tan -1 ENTER Display: 60

3

Using the calculator to find cot :Find cot 257º:

tan x -1257 ENTER ENTER

Display: .2308681911

Method 1

1 tan ENTER

Display: .2308681911

÷ 257Method 2

NOTE: cot tan 1


Trigonometry Basics - Cotangent

• The cotangent of angle B, written cot B, is defined as

cotBC length of the leg adjacent to B

BAC length of the leg opposite B

4

3

5

C B

A


cot B =BCAC

=43

4cot

3

ACA

BC


Meet Chief

Sine - SOH =Opposite

Hypotenuse

Tangent - TOA =Opposite

Adjacent

Cosine - CAH =Adjacent

Hypotenuse

SOH CAH TOA


Trig. Relationships

Recall:

3sin

5

BCA

AB

the sine of an acute angle has the same value as the cosine of its complement.

4

3

5

C B

A

3cos

5

ABB

AC

sin A = cos B and cos A = sin B

The tangent of an acute angle is the reciprocal of the tangent of its complement

tan A · tan B = 1

the tangent of an acute angle has the same value as the cotangent of its complement.

tan A = cot B and cot A = tan B


Model Problem

10

100

6436

86

2

2

222

222

AB

AB

AB

AB

ABACBC

222 bac

TheoremnPythagorea

In right triangle ABC with right angle at C, BC = 6, and AC = 8. Find the three trigonometric functions of B.

8

6B C

A

leg adjacent to B

hypotenuse

leg opposite B

leg adjacent to B

10

leg opposite B

hypotenusesin B =

cos B =

tan B =

8

106

108

6


Model Problem

Park planners would like to build a bridge across a creek. Surveyors have determined that from 5 ft. above the ground the angle of elevation to the top of an 8ft. pole on the opposite side of the creek is 5o. Find the length of the bridge to the nearest foot.

5’ 8’5o

x3’

o 3tan5

x

o

3

tan5x 34.29' 34 feet


Model Problems

1. sin 24o is equivalent to

a) cos 24o b) sin 66o c) cos 660 d) 1/sin 240

2. If cot x = tan(x + 20o), find x.When the cotangent and tangent functions are equal in value, the angles must be complementary.

The sine of an angle has the same value as the cosine of its complement.

x + (x + 20) = 90

2x + 20 = 90

x = 70


Degrees, Minutes & Seconds

3600 in a circle

60 minutes in 1 degree

60 seconds in 1 minute

17o 43’05”

17 degrees 43 minutes 5 seconds

1 minute is 1/60th of a degree

1 second is 1/60th of a minute

o1 4317 43' 17 43 17 17.716

60 60


Model Problem

Find cos 17o 43’ to 4 decimal places

Find sin 20.30o to 4 decimal places

Find sin 20o 30’ to 4 decimal places


Find An Angle Given a Trig Function Value

3cos

2 What is measure of ?

2nd cos -1 ENTER3 22nd ÷

Calculator’s MODE must be in degrees

30o

sin 0.2478

cos 0.2249

tan 0.3987

What is measure of ?


Regents Prep

In triangle ABC, side a = 7, b = 6, and c = 8. Find m B to the nearest degree.

1) 43o 2) 47o

3) 65o 4) 137o


Regents Prep

In the diagram below of right triangle KTW, KW = 6, KT = 5, and mKTW = 90.

5

6

W

T K

What is the measure of K, to the nearest minute?

1) 33o33’ 2) 33o55’

3) 33o34’ 4) 33o56’

aim: the six trigonometric functions course: alg. 2 & trig. aim: what does sohcahtoa have to do...

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