Math-2Lesson 8-7:

Unit 5 Review (Part -2)

Trigonometric Functions

SOH-CAH-TOA

“Some old horse

caught another horse

taking oats away.”

)(

)(sin

lengthhypotenuse

lengthoppositeA

)(

)(cos

lengthhypotenuse

lengthadjacentA

h

aA cos

h

oA sin

)(

)(tan

lengthadjacent

lengthoppositeA a

oA tan

SOH

CAH

TOA

Trigonometric Functions (Mnemonics)

SHA-CHO-CAO

SHA

CHO

CAO

Sine Ratio

7

x

8

hyp

oppA sin

A

B

C

What is the sine

ratio of angle A?

8

7sin A 8

7Sine ratio of angle A is

x 22 78 2285664 8

22sin B

Sine ratio of angle B =

4

2

4

2

3

4

5

A

B

C

What do you notice about

the trig ratios?

4

3tan A

5

3sin A 5

4sin B

5

4cos A

5

3cos B

3

4tan B

BA cossin

Must use trigonometry

Solve the Triangley

x

12

A

Solve for ‘x’ using either:

(1) Pythagorean Theorem (more work)

90 BmAm

B

43º

C9043 Bm47Bm

1243sin

y90Cm

47º

y43sin12

2.8y

y=8.2

(2) Trigonometry (easy)

1243cos

x

x43cos12

8.8x

x=8.8hyp

oppA sin

Could you use to find ‘x’?Bm

Solving a Right Triangle:

43ºA

C

B

47º5

If you don’t know what trig. to use, write equations for all of them

x

y sin 43º = y/5

y = 5 sin 43º = 3.43.4

cos 43º = x/5

x = 5 cos 43º = 3.7

tan 43º = y/x

What if the angle is unknown?

xA

C

B

5

1. Write the trig ratio.

8

8

5sin x

2. To “undo” the sine function, use the

inverse of the sine function.

8

5sin)(sinsin 11 x

8

5sin 1x

7.38x

Angle of Elevation: angle above the horizon that the eye has to

look up to see something.

Angle of Depression: angle below the horizon that the eye has

to look down at something.

The angle of elevation from the buoy to the top of the

Barnegat Bay lighthouse 130 feet above the surface of the

water is 5º. Find the distance x from the base of the

lighthouse to the buoy.

130

x

5º

1. Draw the picture

2. Write the equation.x

130)5tan(

3. Solve for the unknown variable.

ftx 9.1485)5tan(

130

If the height of a building is 470 m and you are standing

100 m away from the building, find the angle of

elevation to the top of the building.

470

100

xº

100

470)tan( x

100

470tan 1x

78x

base of a triangle: any side of a triangle.

How many different ways

are there to calculate

the area of a triangle?

Height = Altitude

three

base = side

height*base*2

1 A

The altitude of a triangle.height*base*

2

1 A

Area formula: requires the use of

matching altitudes and sides.

Using segment AC as the base, requires the use

of segment BD as the height.

Obtuse Triangle: has one angle that is “obtuse”

(measures greater than 90°)

Two of the altitudes will be outside the triangle!

You must extend a side to get a perpendicular intersection.

37º

143º

Areas of Obtuse Triangles: you must solve a right

triangle to find the height.

1. Use linear pair theorem to find the angle opposite the height ‘h’.

737sin

h

2. Use SOHCAHTOA to obtain an equation with ‘h’ in it.

h37sin7

2.4h

3. Find areah*b*0.5 A

)0.5(5)(4.2 A

2units 10.5 A

Find the Area of the Triangle

C B

8

10

A

7.6

48º51º

h

CB

8

10

A

There are two right triangles that can be used to solve for ‘h’.

7.6

48º51º

hh = 8*Sin(48º)8

)48sin(h

hyp

oppA sin

Area = ½(10)(5.9)

h*b*5.0A rea

Area = 29.5 square units

h = 5.9

Area of a Triangle height*base*2

1A rea

C

B

A

5

4

3

For right triangles one of the legs is the base, the other

leg is the height (or vice-versa).

What is the base?

What is the height?

There is no point in finding the height from

Angle C to side AB.

Vocabulary

radius

length measureradian arc

Radian measure: the ratio of the arc length to

the distance the arc is from the vertex of the angle.

Vocabularyradius

length measureradian arc

radius circle a of measure

ncecircumfereradian

Radian measure for a complete circle.

r

r 2 circle a of measure

radian

radians 2 circle a of measure radian

Units of radians = inches/inches

Radian measure has no units! (nice)

Converting from Degrees to Radian Measure

140°

4

3

45*3x

2360

140 xo

Write a proportion, solve for ‘x’.

o135

9

7x

Converting from Radian Measure to degrees.

Write a proportion, solve for ‘x’.

o

x

3602

43

x2*36

142*

18

7

o

x

3602*4

3

x2*4

360*3

2

90*3

Problem types you’ll see:

32

inches 5r

What is length of the subtended arc?

2

angle

ncecircumfere

arc

inches 3

10arc

Write a proportion, solve for ‘x’.

2*3

2

**10

in

arc

2

32

**10

in

arc

50

inches 7r

What is length of the subtended arc?

inches 18

35arc

360

angle

ncecircumfere

arc

Write a proportion, solve for ‘x’.

360

50

7**2

in

arc

inchesarc *7**2*36

5

Sector Area Problems

r = 8 ft

What is the area of the sector?

20º

360

angle

area circle

areasector

Write a proportion, solve for ‘x’.

360

20

)8(* 2

in

area

18

1

*64 2

in

area

18

*64 2inarea

2

9

32in

r = 6.5 ft

Sector area = ?

3

2

Sector Area Problems

2

angle

area circle

areasector

Write a proportion, solve for ‘x’.

2

32

)5.6(* 2

in

area

2*3

2

*7.132 2

in

area

3

*7.132 2inarea

2*2.44 in

45-45-90 Right Triangle

45ºC

B

45º

1

1

A

222 cba

Is the triangle…a) Obtuseb) Isoscelesc) Scalene

Solve for ‘h’ using the

Pythagorean theorem.

222 11 c22 c

2c

2

Label the side lengths of a

similar 45-45-90 triangle using

a scale factor of 3.

45ºC

B

45º

A

233

3

45º

45º

1

12

Use scale factors or proportions to solve for the

lengths of sides of similar 45-45-90 right triangles.

Scale

factor

32

45

45

62

XY

45º

45º

1

12

Solve for ‘x’

Scale

factor 57

45

45

X

107

X

Scale factor: a ratio of the lengths of two

corresponding sides.

2

107

107(SF)*2

57

Building a 30-60-90 triangle.

60º

60º

2

60º

2

2

Start with an “equilateral” triangle whose side lengths are a

convenient number (‘2’).

What are the

measures of the

angles?

60º

2

60º

30º

2

2

1 1

30º

Construct an angle bisector of the top angle.

What lengths are the two segments formed at the

bottom of the original triangle?

CPCTC (bottom legs

are congruent so each

is ½ the total bottom

length.

Why?

2x

1

30º

60º

We now have a 30-60-90 triangle.

222 cba

222 21 a

142 a

32 a3x

2

You have to remember this triangle.

1. The shortest side is opposite the smallest angle. 2

1

30º

60º

3

2. The longest side is opposite the largest angle.

1 2

Which is a larger number; 2 or ?3

1 3 4

72.1

c

a

30º

60º

We use similarity to solve

for the missing sides.

2

6

Scale

factor

2

6)(*3 SF

2

1

30º

60º

3

What are the

measures ‘c’ and ‘a’ ?

32

6)( SF

32

23)( SF

2

2)( SF

2

2a 2c