Section 1.2Trigonometric Ratios

Section 1.2Trigonometric Ratios

Objectives:

1. To state and apply the Pythagorean theorem.

2. To define the six trigonometric ratios.

Objectives:

1. To state and apply the Pythagorean theorem.

2. To define the six trigonometric ratios.

Pythagorean TheoremIn right ABC, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2

BB

AA

CC

hypotenusehypotenuseleg opposite Aleg opposite A

sine of A =sine of A =

hypotenusehypotenuseleg adjacent Aleg adjacent A

cosine of A =cosine of A =

leg adjacent Aleg adjacent Aleg opposite Aleg opposite A

tangent of A =tangent of A =

Trigonometric Ratios

SOHCAHTOA

SOHCAHTOA

ineppositeypotenuseosinedjacentypotenuseangentppositedjacent

hhoo

AAsinsin ==hhaa

AAcoscos ==aaoo

AAtantan ==

A

opposite

adjacent

hypotenuse

Trigonometric RatiosTrigonometric Ratios

cosAcosA11

secant of A = secA = secant of A = secA =

sinAsinA11

cosecant of A = cscA = cosecant of A = cscA =

tanAtanA11

cotangent of A = cotA = cotangent of A = cotA =

Reciprocal RatiosReciprocal Ratios

oohh

AAcsccsc ==aahh

AAsecsec ==ooaa

AAcotcot ==

Reciprocal RatiosReciprocal Ratios

A

opposite

adjacent

hypotenuse

EXAMPLE 1 Find the six trigonometric ratios for G in right EFG.

EXAMPLE 1 Find the six trigonometric ratios for G in right EFG.

6688

GG

EE FF

g2 + e2 = f2

g2 + 62 = 82

g2 + 36 = 64g2 = 28g = 2 7

Practice Question: Find the six trigonometric ratios for E in right EFG.

Practice Question: Find the six trigonometric ratios for E in right EFG.

991111

GG

EE FF

sin E =

1. 2.

3. 4.

911

2 1011

9 1020

2 109

yy

xx

rr

P(x,y)P(x,y)

yyxx

oppoppadjadj

cotcot ====xxyy

adjadjoppopp

tantan ====

xxrr

adjadjhyphyp

secsec ====rrxx

hyphypadjadj

coscos ====

yyrr

oppopphyphyp

csccsc ====rryy

hyphypoppopp

sinsin ====

Trigonometric RatiosTrigonometric Ratios

EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.

PP90°90°

P = (0, 1)x=0, y=1, r=1cos = 0sin = 1tan = und.

EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.

PP90°90°

P = (0, 1)x=0, y=1, r=1sec = und.csc = 1cot = 0

Practice Question: Find the six trigonometric ratios for a 180º angle.Practice Question: Find the six trigonometric ratios for a 180º angle.

PP

180°180°

sin = 1. -1 2. 03. 1 4. und.

Special TrianglesSpecial Triangles

22cc ==

4545

11

45451

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

c2 = 2

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

c2 = 2

2

1

Special TrianglesSpecial Triangles

22cc ==

4545

11

45451

sin 45° =2

2

22

Special TrianglesSpecial Triangles

22cc ==

4545

11

45451

sin 45° =

cos 45° =

tan 45° =

22

1

2

Special TrianglesSpecial Triangles

22cc ==

4545

11

45451

csc 45° =

sec 45° =

cot 45° =

2

1

2222

sin 45° =sin 45° =

cos 45° =cos 45° =

tan 45° =tan 45° =

2222

11

22csc 45° =csc 45° =

sec 45° =sec 45° =

cot 45° =cot 45° =

22

11

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

a2 + b2 = c2

12 + b2 = 22

1 + b2 = 4b2 = 3

a2 + b2 = c2

12 + b2 = 22

1 + b2 = 4b2 = 3

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

tan 30° =3

1

3

3

3333

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

tan 30° =

sin 30° =

cos 30° =2233

2211

33

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

cot 30° =

csc 30° =

sec 30° =

22

333322

3333

tan 30° =

sin 30° =

cos 30° =2233

2211

33cot 30° =

csc 30° =

sec 30° =

22

333322

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

sin 60° =

cos 60° =

tan 60° =

2233

2211

33

3333

33bb ==

Special TrianglesSpecial Triangles

6060

22

11

3030

csc 60° =

sec 60° =

cot 60° =

333322

22

sin 60° =

cos 60° =

tan 60° =

2233

2211

333333

csc 60° =

sec 60° =

cot 60° =

333322

22

Homework:

pp. 12-13

Homework:

pp. 12-13

►A. Exercises1. Find the six trig. ratios for both

acute angles in each triangle.

►A. Exercises1. Find the six trig. ratios for both

acute angles in each triangle.

5

C

A

B

2

21

12

LM

N2

1st-solve for side n1st-solve for side n22 + n2 = 12222 + n2 = 122n2 = 122-22 = 144-4 = 140n2 = 122-22 = 144-4 = 140

n

►A. Exercises3. Find the six trig. ratios for both

acute angles in each triangle.

►A. Exercises3. Find the six trig. ratios for both

acute angles in each triangle.

353522140140nn ====

#3.#3.

12

LM

N2

352

►A. Exercises7. Find the six trig. functions for an

angle in standard pos. whose terminal ray passes through the point (-6, -1).

►A. Exercises7. Find the six trig. functions for an

angle in standard pos. whose terminal ray passes through the point (-6, -1).

-6-6-1-1

3737

-6-6-1-1

3737

sin = csc =

cos = sec =

tan = cot =

►B. Exercises15. Find the six trig. ratios for the

quadrantal angle measuring 180°.

►B. Exercises15. Find the six trig. ratios for the

quadrantal angle measuring 180°.

(-1, 0)(-1, 0)

►B. Exercises15. Find the six trig. ratios for the

quadrantal angle measuring 180°.

►B. Exercises15. Find the six trig. ratios for the

quadrantal angle measuring 180°.

■ Cumulative Review30. Give the distance between (2, 7) and

(-3, -1).

■ Cumulative Review30. Give the distance between (2, 7) and

(-3, -1).

■ Cumulative Review31. Give the midpoint of the segment

joining (2, 7) and (-3, -1).

■ Cumulative Review31. Give the midpoint of the segment

joining (2, 7) and (-3, -1).

■ Cumulative Review32. Give the angle coterminal with 835

if 0 360.

■ Cumulative Review32. Give the angle coterminal with 835

if 0 360.

■ Cumulative Review33. Convert 88 to radians.■ Cumulative Review33. Convert 88 to radians.

■ Cumulative Review34. If sec = 7, find cos .■ Cumulative Review34. If sec = 7, find cos .