5.2 TRIGONOMETRIC RATIOS IN RIGHT TRIANGLES• WHAT ARE THE SIX TRIGONOMETRIC RATIOS IN A RIGHT

TRIANGLE?

• HOW DO WE FIND THE VALUES OF THE SIX TRIG RATIOS?

WHAT IS A TRIG RATIO?

A trigonometric ratio is a ratio of two sides of a right triangle in relation to a specific acute angle. Each trig ration has a specific name to distinguish what sides are being used.

is read as “the sine of alpha is 3/5”

-NOTE is read “the square root of a”, you have to tell what you are taking the square root OF, just like you have to tell what ANGLE you are using for the ratio.

WHAT DO WE NEED TO KNOW ABOUT TRIG RATIOS?

The trig ratios are instructions on how to find a particular number, in order to do that you need to know some background information.

1. Pythagorean theorem

• a, b are legs of a right triangle (across from acute angles)• c is the hypotenuse (longest side, across from right angle)

2. Opposite/Adjacent/Hypotenuse

• The side “opposite” depends on the angle, if you change the angle you change the side that is “opposite”, similarly with “adjacent.”

• What side is “opposite” to C?

EXAMPLE 1: FIND THE MISSING SIDE OF EACH TRIANGLE

A.

B.

C.

D.

E.

F.

EXAMPLE 2: IDENTIFY THE OPPOSITE, ADJACENT, AND HYPOTENUSE TO THE GIVEN ANGLE

• Angle A

• Opposite = 4

• Adjacent =3

• Hypotenuse = 5

• Angle C

• Opposite = 3

• Adjacent =4

• Hypotenuse = 5

• Angle B

• Opposite = 5

• Adjacent =Not well defined ??

• Hypotenuse = • 5

WHAT ARE THE SIX TRIG RATIOS?

Original Trig Ratios

Words Symbols Ratio

Sine of

Cosine of

Tangent of

Reciprocal Trig Ratios

Words Symbols Reciprocal Ratio

Cosecant of

Secant of

Cotangent of

EXAMPLE 3: GIVEN ONE TRIG RATIO, FIND THE OTHER 5 FOR THE GIVEN ANGLE

Note: in order to find all the trig ratios we must first know ALL the sides and then be able to identify the opposite, adjacent and hypotenuse. It is often helpful to sketch a picture.

A.

𝛽4

53H

A

O

53

35

34

45

43

• Find all the sides

• Identify O, A, H

EXAMPLE 3: GIVEN ONE TRIG RATIO, FIND THE OTHER 5 FOR THE GIVEN ANGLE

Note: in order to find all the trig ratios we must first know ALL the sides and then be able to identify the opposite, adjacent and hypotenuse. It is often helpful to sketch a picture.

B.

𝛽

15

178H

O

A1517

• Find all the sides

• Identify O, A, H

1581715178815

EXAMPLE 4: FIND THE TRIG RATIOS FOR THE SPECIAL RIGHT TRIANGLES

A.

A

O

• Identify O, A, H• Does it matter

what we call A or O?

H𝑥2𝑥

=12

𝑥 √32𝑥

=√32

𝑥 √3𝑥

=√31

=√3

𝑥𝑥 √3

= 1√3

=√33

2𝑥𝑥

=2

2𝑥𝑥 √3

= 2√3

=2√33

EXAMPLE 4: FIND THE TRIG RATIOS FOR THE SPECIAL RIGHT TRIANGLES

B. H

O

A

• Identify O, A, H• Does it matter

what we call A or O?

𝑥𝑥

=1

𝑥 √2𝑥

=√2

𝑥𝑥 √2

= 1√2

=√22

𝑥𝑥 √2

= 1√2

=√22

𝑥𝑥

=1

𝑥 √2𝑥

=√2

EXAMPLE 4: FIND THE TRIG RATIOS FOR THE SPECIAL RIGHT TRIANGLES

C.

O

A

• Identify O, A, H• Does it matter

what we call A or O?

H

𝑥2𝑥

=12

𝑥 √32𝑥

=√32

𝑥 √3𝑥

=√31

=√3

𝑥𝑥 √3

= 1√3

=√33

2𝑥𝑥

=2

2𝑥𝑥 √3

= 2√3

=2√33

SUMMARY

1. Find the six trig ratios (of ) for a triangle with the side opposite measuring 6, and the hypotenuse of the triangle measuring 12.

2. If , find .

3. Given triangle ABC, with right angle B, which trig ratio could be used to find the length of ?

A.

B.

C.

D.

SUMMARY1. Find the six trig ratios (of ) for a triangle with the side opposite

measuring 6, and the hypotenuse of the triangle measuring 12.

• Using Pythagorean theorem the last side is which makes this a triangle so see example 4C.

2. If , find .

• Find the 3rd side using Pythagorean theorem, which will be 5. See example 3A.

3. Given triangle ABC, with right angle B, which trig ratio could be used to find the length of ?

A.

B.

C.

D. • DRAW A PICTURE

A

CB