Topic 1 Pythagorean Theorem and SOH CAH TOA Unit 3 Topic 1.

Topic 1Pythagorean Theorem and SOH CAH TOA

Unit 3 Topic 1

Before we start…• Trigonometry means triangle measurement.

• Trigonometry is used to measure distances that are difficult to measure directly.▫ the distance across a canyon, the height of a

mountain, how to land an airplane, or the length of a tunnel

• Trigonometry is used to calculate unknown measures when you know:▫ the length of two sides▫ the length of one side and the measure of one

angle

Before we start…

• The triangle to the right is called .

• Triangles have 3 angles and 3 sides.▫ We label the vertices with CAPITAL LETTERS.▫ We label the sides with lower-case letters.▫ Side c is the LONGEST side, across from the right

angle (90˚) and is called the hypotenuse.▫ Sides a and b are the legs of the right triangle and

are called the adjacent and opposite sides.

• The sum of the angles in any triangle is 180.

ABC

A

BCθ

b(opposite)

a(adjacent)

c(hypotenuse)

Before we start…

Label: a) the sides, using b) the adjacent side, the

opposite side, lower-case letters. and the hypotenuse, given reference B.

A

B

C

θ

E

GF

Before we start…

Label: a) the sides, using b) the adjacent side, the

opposite side, lower-case letters. and the hypotenuse, given reference B.

A

B

C

θ

E

GF e

g f

The lower-case letters are written across from their upper case counterparts.

opposite

hypotenuseadjacent

The hypotenuse is across from the right angle, the opposite side is across from the reference angle, and the opposite side is the one left over.

Information

• When two side lengths of a right triangle are known, we can use the Pythagorean Theorem, , to solve for the length of the missing side.

2 2 2 a b c

Substitute known values into the formula, keeping in mind that the c value is reserved for the hypotenuse.

Example 1Determining the length using Pythagorean Theorem

Determine the missing lengths for the following triangles. Round to the nearest tenth. (a-d).

a)

Try this on your own first!!!!

A

BC

5 cm

8 cm

Solution:

2 2 2

2 2 2

2

5 8

89

9.4

a b c

c

c

c cm Take the square root of both sides.

The c value is reserved for the hypotenuse.



b)


Solution:

2 2 2

2 2 2

2

12 22.3

641.29...

25.3

a b c

c

c

c cm

A

B

C

12 m 22.3 m

Substitute known values into the formula, keeping in mind that the hypotenuse (17) must be put into the formula as c.



c)


Solution:2 2 2

2 2 2

2

2

2

8 17

64 289

225

15

a b c

b

b

b

b

Take the square root of both sides.

A

CB

178

Subtract 64 from both sides.

The hypotenuse, 32, must be put into the formula as c.



a)


Solution:2 2 2

2 2 2

2

2

20 32

400 1024

624

25.0

a b c

b

b

b

b units

3220

A

B

C

Example 2Determine the angle

Determine the missing angles in the following triangles.

a) b) c)


45

x62

x

80

25

30x

Example 2Determine the angle

Remember: All the angles in a triangle must add up to 180⁰. When we are given 2 of the 3 angles, we can subtract them both from 180⁰ to see what we will have left for the third.

a) b) c)


45

x 62

x

80

25

30x

180 45 90 45

45

x

180 80 62 38

38

x

180 30 25 125

125

x

Information

• Sometimes we are only given one angle and one side in a right triangle and asked to find a missing angle or side.

• How do we do this? Can we use Pythagorean

Theorem? Do you remember SOH CAH TOA?

Information• SOH CAH TOA is the word that we use to

remember what to do with right triangles to solve for either a missing angle (when given two sides), or a missing side (when given one angle and one side).

A

BCθ

Information

• SOH CAH TOA is the word that we use to remember these ratios.

InformationTo find a missing side using the trig ratios:

• Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.

• Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side.

• Substitute all known values into the equation, including the variable for the unknown side length.

• Cross multiply and divide to determine the unknown side length.

Example 3

Determine the length of the missing side. Round all answers to the nearest tenth.

a) b) c)


20

52 cmx

4719 m

x 12 m

53

x

cos( )

cos(20 )52

adjacent

hypotenuse

x

cm

cos( )

cos(2

52cos(20 )

4

0 )52

8.9

adjacent

hypotenuse

x

xm

cm

c

x

Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.

Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side.

Substitute all known values into the equation, including the variable for the unknown side length.

Cross multiply and divide to determine the unknown side length.

Example 3: Solution

20

52 cm x

a)

opposite

hypotenuse

adjacent

Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle.

Remember: Since the missing side is the adjacent side, and a given side is the hypotenuse, we select the cosine ratio.

4719 m

x

tan( )

tan(47 )19

opposite

adjacent

x

cm

tan( )

tan(47 )

19 tan(47 )

20.4

19

opposite

adjacent

x

x

m

m

x





Example 3: Solution

b)

opposite

hypotenuse

adjacent


Remember: Since the missing side is the opposite side, and a given side is the adjacent, we select the tangent ratio.

sin( )

12sin(

sin(53 ) 12

15.

3

0

5 )

opposite

hypotenuse

x

m

x

x

12 m

53

x

sin( )

12sin(53 )

opposite

hypotenuse

m

x





Example 3: Solution

c)

opposite hypotenus

e

adjacent


Remember: Since the missing side is the hypotenuse, and a given side is the opposite, we select the sine ratio.

Example 4Determining the length

A ladder resting on the side of a building forms an angle of 44 with the ground. If the ladder is touching the building at a height of 8 m from the ground, how long is the ladder, to the nearest tenth?


Example 4: Solution

oppositehypotenuse

mx

x

x m

sin

8sin44

sin44 8

811.5

sin44

44⁰

Draw a triangle and label what you know and what you are going to solve for.

x8m

hypotenuse

opposite

Based on the reference angle, we have the opposite side and the hypotenuse. This means that we can use the sine ratio.

Information

To find a missing angle using the trig ratios:

• Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle.

• Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides.

• Substitute all known values into the equation, including the variable for the unknown angle measure.

• Use the inverse function (sin-1, cos-1, or tan-1) to find the angle.

Example 5Determining the angle

Determine the measure of the missing angle. Round all answers to the nearest degree (whole number).

a) b) c)


x

10 m

8 m 20 cm

30 cm

x

x

2 m

5 m

tan( )

8tan( )

10

opposite

adjacent

mx

m

1

tan( )

8tan( )

8tan 3

0

1

91

0

opposite

adjacent

mx

m

m

m

Example 5: Solution

a)x

10 m

8 m

Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle.

Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides.

Substitute all known values into the equation, including the variable for the unknown angle measure.

Use the inverse function (sin-1, cos-1, or tan-1) to find the angle.

opposite

hypotenuseadjacent


Remember: Since the two sides we are given are opposite and adjacent, we can use the tangent ratio.

1

cos( )

20cos( )

3020

cos 4830

adjace

cm

cm

nt

hypotenuse

cmx

cm

cos( )

20cos( )

30

adjacent

hypotenuse

cmx

cm

Example 5: Solution

b)





opposite

hypotenuse

adjacent


Remember: Since the two sides we are given are adjacent and hypotenuse, we can use the cosine ratio.

20 cm

30 cm

x

1

sin( )

2sin( )

22

5

5

sin 4

opposite

hypotenuse

mx

mm

m

sin( )

2sin( )

5

opposite

hypotenuse

mx

m

Example 5: Solution

c)





opposite

hypotenuse

adjacent


Remember: Since the two sides we are given are opposite and hypotenuse, we can use the sine ratio.

x

2 m

5 m

Example 6Determining the Angle

A 56 m wire, supporting a TV tower 45 m tall, joins the top of the tower to an anchor point on the ground. What is the angle, , to the nearest degree, that the wire makes with the ground?


Example 6: Solution

oppositehypotenuse

mm

mm

1

sin

45sin

56

45sin 53

56

opposite

hypotenuse

Based on the reference angle, we have the opposite side and the hypotenuse. This means that we can use the sine ratio.

Need to Know:Solving for a missing side in a right triangle:• When two sides of a right triangle are known, we can

use the Pythagorean Theorem, to determine the third side.

• If an angle and only one side are known:▫Label the sides as opposite, adjacent and

hypotenuse, with respect to the given angle.▫Determine which trig ratio (sin, cos, or tan) includes

the given angle, the given side, and the missing side. ▫Substitute all known values into the equation,

including the variable for the unknown side length.▫Cross multiply and divide to determine the unknown

side length.

Need to Know:Solving for a missing angle in a right triangle:• When two angles of a right triangle are known, we

can use the triangle sum theorem to find the third angle.

• If only sides are given:▫ Label the sides as opposite, adjacent, and

hypotenuse, with respect to the unknown angle. ▫ Determine which trig ratio (sin, cos, or tan) would

include the unknown angle and the two given sides.▫ Substitute all known values into the equation,

including the variable for the unknown angle measure.▫ Use the inverse function (sin-1, cos-1, or tan-1) to find

the angle. You’re ready! Try the homework from this section.

Topic 1 Pythagorean Theorem and SOH CAH TOA Unit 3 Topic 1.

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