Quiz 13-2

1.

2.

3.

?3

2

?20 o

Convert to degreesConvert to degrees

Convert to radiansConvert to radians

Arc length = Arc length = inches 3

2

Radius = Radius = 6 inches

What is the angle measure (in radians)?What is the angle measure (in radians)?

rs

13-1 Trigonometric Functions of Acute

Angles

What you’ll learn about

• Right Triangle Trigonometry• Two Famous Triangles• Evaluating Trigonometric Functions with a

Calculator• Applications of Right Triangle Trigonometry

… and whyThe many applications of right triangle

trigonometry gave the subject its name.

Right Triangle Review

What is the measure What is the measure of this angle?of this angle?

9090ºº

What measure do these What measure do these two angles add up to?two angles add up to?AA

BBCC

90 BmAm

Right Triangle Review

What are these two sideWhat are these two side of the triangle called?of the triangle called?

What is the What is the namename of this of this side of the triangle?side of the triangle?AA

BBCC

hypotenusehypotenuse

legslegs

Right Triangle ReviewThe The Pythagorean TheoremPythagorean Theorem relates the lengths relates the lengths of the two of the two legslegs to the to the hypotenusehypotenuse. What is the. What is the equation of this relation?equation of this relation?

aa

bb

cc222 cba

Right Triangle Review

If: a = 3 b = 4 If: a = 3 b = 4 Then: c = ?Then: c = ?

aa

bb

cc

222 cba

33

44

(formula problem)(formula problem)

222 43 c25169 c

c = 5c = 5

Right Triangle Review

If: c = 7 b = 4 If: c = 7 b = 4 Then: a = ?Then: a = ?

aa

bb

cc

222 cba

1010

44

(formula problem)(formula problem)222 74 a

331649 a

222 47 a

Your turn:

1. 1.

aa

bb

cc

2

2a

2

2b

c = ?c = ?

2. 2. 1c

222 cba

2

3b

a = ?a = ?

Trigonometric Functions

SOHCAHTOASOHCAHTOA

““SSome ome oold ld hhorse orse ccaught aught aanother nother hhorse orse ttaking aking ooats ats aaway.”way.”

)(

)(sin

lengthhypotenuse

lengthoppositeA

Key pointKey point: sine : sine of an angleof an angle (measured in degrees or radians)(measured in degrees or radians)

)(

)(cos

lengthhypotenuse

lengthadjacentA

h

aAcos

h

oAsin

)(

)(tan

lengthadjacent

lengthoppositeA

a

oAtan

These only work for These only work for rightright triangles!!! triangles!!!

Sine Ratio33

xx

55

hyp

oppAsin

AA

BB

CC

What is the sine What is the sine ratio for angle A?ratio for angle A?

5

3sin A

oppopp

5

3Sine ratio for angle A isSine ratio for angle A is

Sine ratio for angle B = ?Sine ratio for angle B = ?

x 22 35 416925 5

4sin B

Sine ratio for angle B =Sine ratio for angle B =5

4

Your Turn:88

xx

1717

hyp

oppAsin

AA

BB

CC

3. What is the sine ratio for 3. What is the sine ratio for angle Aangle A??17

8

4. What is the sine ratio for 4. What is the sine ratio for angle Bangle B??17

15

Cosine Ratio33

44

55

hyp

adjAcos

AA

BB

CC

What is the cosine What is the cosine ratio for angle A?ratio for angle A?

5

4cos A

adjadj

5

4Cosine ratio for angle A isCosine ratio for angle A is

Your Turn:88

1515

1717

AA

BB

CC

3. What is the cosine ratio for 3. What is the cosine ratio for angle Aangle A?? 17

15

4. What is the cosine ratio for 4. What is the cosine ratio for angle Bangle B??17

8

hyp

adjAcos

Tangent Ratio33

44

55

adj

oppAtan

AA

BB

CC

What is the tangent What is the tangent ratio for angle A?ratio for angle A?

4

3tan A

adjadj

4

3Tangent ratio for angle A isTangent ratio for angle A is

opp

Your Turn:88

1515

1717

adj

oppAtan

AA

BB

CC

5. What is the tangent ratio for 5. What is the tangent ratio for angle Aangle A??15

8

6. What is the tangent ratio for 6. What is the tangent ratio for angle Bangle B??8

15

Evaluating Trig. Functions of 45º

4545ºº

xx = ?x = ? x = 45x = 45ºº

AA

CC

BB

If AB = If AB =

4545ºº

AC = ?AC = ?

11AC

22

BC = ?? BC = ??

22

22

22

22

22

22

AC

222 cba 42

42 AC

Your Turn:

Find all 3 trigonometric ratios of a 45º – 45º – 90º triangle. 4545ºº

AA

CC

BB

4545ºº22

22

1

tansin

45 7. 8. 9.7. 8. 9.

cos

What if the triangle is still 45-45-90 BUT is bigger?

FindFind: sin 45: sin 45ºº2

1

2

2

2

2*

Sin 45Sin 45º does not change regardless º does not change regardless of the size of the 45-45-90 triangle!!!of the size of the 45-45-90 triangle!!!

Rationalize the denominatorRationalize the denominator

Which one is easier to calculate: Which one is easier to calculate:

2 (1) hypotenuse = (1) hypotenuse =

(2) hypotenuse = (2) hypotenuse = 1

That’s why we use the UNIT CIRCLE, the hypotenuse of the triangle is

always ‘1’.

r = 1

45º

22

22

FindFind: sin 45: sin 45º = º = 2

2 Let’s put the triangleLet’s put the triangle on top ofon top of the unit circle. the unit circle.

45º

Trig. Functions of the 30º-60º-90º triangle

3030ººAA

CC

BB

26060ºº

1

3

tansin

30

cos

60

AA

CC

BB

1½

23

Shrink Shrink

by by ½½

3030ºº

6060ºº

½½

23

2321

30tan

3

2*2

1

3

3

3

3*3

1

33

23

½½ 3

8.8.

9.9.

Your Turn:Your Turn:

Use your calculator to find trig ratios

Sin 30Sin 30º = ?º = ? Make sure it’s in Make sure it’s in degreedegree mode. mode.

Sin 30Sin 30º = º = 0.50.5

Sin 60Sin 60º = ?º = ?

tansin

30

cos

60

½½

23

33

23

½½ 3

Sin 60Sin 60º = º = 0.8660 238660.0

Relate back to the UNIT CIRCLE

r = 1FindFind: sin 60: sin 60ºº

23

123

Using the unit circle:Using the unit circle:

Using the upper right triangle:Using the upper right triangle:

3

3030ºº

AA

CC

6060ºº

BB

3030ººAA

CC

BB

26060ºº

1

3

23

2

1

2 23

Solving a Right Triangle: Find the measure of

every other angle and side.

A right triangle has a hypotenuse that is 5 A right triangle has a hypotenuse that is 5 inches long. One of the acute angles is 43inches long. One of the acute angles is 43º.º.

4343ººAA

CC

BB

1.1. Draw the picture with Draw the picture with the given information.the given information.

2. Pick the “low hanging 2. Pick the “low hanging fruit” (do the easy parts) fruit” (do the easy parts)

5757ºº5

3. Use trig ratios to solve for the unknown side lengths.3. Use trig ratios to solve for the unknown side lengths.x

y

sin 43sin 43º = y/5º = y/5 y = 5 sin 43º = y = 5 sin 43º = 3.43.4

3.43.4

cos 43cos 43º = x/5º = x/5 x = 5 cos 43º = x = 5 cos 43º = 3.73.7

Your Turn:

2525ººAA

CC

BB

10. 10.

11. 11.

12. 12.

22?Cm

AB = ?AB = ?

BC = ?BC = ?

VocabularyAngle ofAngle of elevation elevation: angle above horizon.: angle above horizon.

Horizontal line Horizontal line

Angle of Angle of depressiondepression: angle below horizon.: angle below horizon.

Horizontal line Horizontal line

HOMEWORK

Section 13-1 (page 856)

(evens) (3 trig functions not 6) 4 (3 pts), 10 (2 pts),

18-20, (1 pt each),

22-28 (3 pts each),

32 (2 pts)

Review: 44 (1 pt)

(25 points)