Trigonometric Ratios

Lesson 12.1

HW: 12.1/1-22

Find the missing measures. Write all answers in radical form.

60°

30° 10

y

z

Warm – up

3

45y60

30x

45

23

33

y

x

5

35

y

z

What is a trigonometric ratio?

The relationships between the angles and the sides of a right triangle are

expressed in terms of TRIGONOMETRIC RATIOS.

We need to do some housekeeping before we

can proceed…

In trigonometry, the ratio we are talking about is the comparison of the sides of a

RIGHT TRIANGLE.

Several things MUST BE understood:1.This is the hypotenuse.. This will ALWAYS be the hypotenuse2.This is 90°… this makes the right triangle a right triangle….

One more thing…

The 2 other angles and the 2 other sides

A

We will refer to the sides in terms of

their proximity to the angle

If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course

we have the hypotenuse.

opposite

adja

cen

t hypotenuse

hypotenuse

Ad

jace

nt

sid

e

Opposite side

Greek letter ‘PHI’

B

If we look at angle B, there is one side that is adjacent to it and the other side is opposite

from it, and of course we have the hypotenuse.

op

po

site

adjacent

hypotenuse

Adjacent side

Op

pos

ite

sid

e

hypotenuse Greek Letter

‘Theta’

Remember we won’t use the right angle

X

Here we go!!!!

The Trigonometric Functions we will be

looking at

SINE

COSINE

TANGENT

The Trigonometric Functions

SINE

COSINE

TANGENT

SINE

Pronounced“sign”

sin

Pronounced

COSINE

“co-sign”

cos

Pronounced

TANGENT

“tan-gent”

tan

The Trigonometric Ratios

So, what does this stuff mean?...

opp

osite

hypotenuse

SinOpp

Hyp

Leg

adjacent

CosAdj

Hyp

Leg

TanOpp

Adj

Leg

Leg

hypotenuse

opp

osite

adjacent

We need a way to remember all of these ratios…

Old Hippie

Old Hippie

SomeOldHippieCameAHoppin’ThroughOurApartment

Old Hippie

Old Hippie

SinOppHypCosAdjHypTanOppAdj

SOHCAHTOA

Definition of Sine Ratio

For any right-angled triangle

sin = opposite side

hypotenuse

Definition of Cosine Ratio

For any right-angled triangle

cos = hypotenuse

adjacent side

Definition of Tangent Ratio.

For any right-angled triangle

tan = opposite side

adjacent

Find:

Sin 16°

Tan 58°

Ex. 1: Finding Trig Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacent

8

17≈ 0.4706

15

17≈ 0.8824

8

15

≈ 0.5333

4

8.5≈ 0.4706

7.5

8.5≈ 0.8824

4

7.5≈ 0.5333

Trig ratios are often expressed as decimal approximations.

SohC

ahT

oa

Ex. 2: Finding Trig Ratios

<S

sin S = opphyp

cos S = adj

hyp

tan S = opp adj

513 ≈0.3846

1213

≈0.9231

5

12≈0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

SohCahToa

Finding sin, cos, and tan.

(Just writing a ratio or decimal.)

Find the sine, the cosine, and the tangent of angle A.

Give a fraction and decimal answer (round to 4 places).

hyp

oppA sin

8.10

9 8333.

hyp

adjA cos

8.10

6 5556.

adj

oppA tan

6

9 5.1

9

6

10.8

A

Shrink yourself down and stand where the angle is.

Now, figure out your ratios.

Find the sine, the cosine, and the tangent of angle A

A

24.5

23.1

8.2

hyp

oppA sin

5.24

2.8 3347.

hyp

adjA cos

5.24

1.23 9429.

adj

oppA tan

1.23

2.8 3550.

Give a fraction and decimal answer (round to 4 decimal places).

Shrink yourself down and stand where the angle is.Now, figure out your ratios.

Finding an angle.(Figuring out which ratio to use and getting

to use the 2nd button and one of the trig buttons.)

Ex. 1: Find . Round to four decimal places.

9

17.2

Make sure you are in degree mode (not radians).

9

2.17tan

2nd tan 17.2 9

3789.62

)

Shrink yourself down and stand where the angle is.Now, figure out which trig ratio you have and set up the problem.

SohCahToa

Ex. 2: Find . Round to three decimal places.

23

7

Make sure you are in degree mode (not radians).

23

7cos

2nd cos 7 23

281.72

)

SohCahToa

Finding the angle measure

4

7

In the figure, find

sin = opposite side

hypotenuse

=4

7

= 34.85

sin

Finding the length of a side

11

In the figure, find y

sin 35 = opposite sidehypotenuse

y

11

y = 6.31

35°35°

y

sin 35 =

11* sin35 = y

3

8

In the figure, find

cos = adjacent Side

hypotenuse

= 38

cos =

= 67.98

Finding the angle measure

6

In the figure, find x

cos 42 = adjacent side

hypotenuse

6x

x = 8.07

42°42°

xcos 42 =

x =6

cos 42

Finding the length of a side

3

5

In the figure, find

tan = adjacent side

opposite side

=35

tan =

= 78.69

Finding the angle measure

z

5

In the figure, find z

tan 22 = adjacent Side

Opposite side

5z

z = 12.38

2222

tan 22 =

5

tan 22z =

Finding the length of a side

Conclusion

hypotenuse

side oppositesin

hypotenuse

sidedjacent acos

sidedjacent a

side oppositetan

Make Sure that the triangle is right-angled

SohCahToa

Solving a Problem withthe Tangent Ratio

60º

53 ft

h = ?

We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:

ft 92353

60tan5353

60tan

5360tan

h

h

h

h

adj

opp

SohCahToa

Note:

• The value of a trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

2

1

Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30

30

sin 30= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

√3

cos 30=

tan 30=

√32

≈ 0.8660

12 = 0.5

√33 ≈ 0.5774

Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2.

30

√31 =

Ex: 5 Using a Calculator

• You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

Sample keystrokesSample keystroke

sequencesSample calculator display Rounded

Approximation

74

or 74 0.961262695 0.9613

or

0.275637355 0.2756

3.487414444 3.4874

sinsin

ENTER

74

74

COS

COS

ENTER

74

or 74

TAN

TANENTER

Notes:• If you look back at Examples 1-5, you

will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.