8.4 Trigonometric Ratios- Sine and Cosine · trigonometric ratios involve the ratio of a leg of a...

8.4 Trigonometric

Ratios- Sine and

Cosine Geometry

Mr. Peebles

Spring 2013

Daily Learning Target (DLT)

Wednesday April 17, 2013 • “I can apply my knowledge of right triangles

to find the sine and cosine of an acute angle.”

Trigonometric Ratios

• Let ∆ABC be a right

triangle. The since,

the cosine, and the

tangent of the

acute angle A are

defined as follows.

ac

bside adjacent to angle A

Side

opposite

angle A

hypotenuse

A

B

C

sin A = Side opposite A

hypotenuse

= a

c

cos A = Side adjacent to A

hypotenuse

= b

c

tan A = Side opposite A

Side adjacent to A

= a

b

Trigonometric Ratios

• TOA Tangent = Opposite/Adjacent

• CAH Cosine = Adjacent/Hypotenuse

• SOH Sine= Opposite/Hypotenuse

Trigonometric Ratios

• TOA “Together Only Actors

• SOH Sing On Holidays

• CAH Cheering All Happily.”

Ex. 1: Finding Trig Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small

sin A = opposite

hypotenuse

cosA = adjacent

hypotenuse

Trig ratios are often expressed as decimal approximations.

Ex. 1: Finding Trig Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small

sin A = opposite

hypotenuse

cosA = adjacent

hypotenuse

8

17 ≈ 0.4706

15

17 ≈ 0.8824

4

8.5 ≈ 0.4706

7.5

8.5 ≈ 0.8824

Trig ratios are often expressed as decimal approximations.

Ex. 2: Finding Trig Ratios

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

5

13 ≈ 0.3846

12

13 ≈ 0.9231

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of the

indicated angle.

R

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

adjacent

opposite12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of the

indicated angle.

R

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

12

13 ≈ 0.9231

5

13 ≈ 0.3846

adjacent

opposite12

13 hypotenuse5

R

T S

Ex. 3: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of 45

45

sin 45= opposite

hypotenuse

adjacent

hypotenuse

1

hypotenuse1

√2

cos 45=

Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2.

45

Ex. 3: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of 45

45

sin 45= opposite

hypotenuse

adjacent

hypotenuse

1

hypotenuse1

√2

cos 45= 1

√2 =

√2

2 ≈ 0.7071

1

√2 =

√2

2 ≈ 0.7071

Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2. 45

2

1

Ex. 4: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of 30

30

sin 30= opposite

hypotenuse

adjacent

hypotenuse

√3

cos 30=

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

2

1

Ex. 4: Finding Trig Ratios—Find the sine,

the cosine, and the tangent of 30

30

sin 30= opposite

hypotenuse

adjacent

hypotenuse

√3

cos 30= √3

2 ≈ 0.8660

1

2 = 0.5

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

Sample keystrokes Sample keystroke

sequences

Sample calculator display Rounded

Approximation

0.275637355 0.2756 sin

sin

ENTER

COS

Sample keystrokes Sample keystroke

sequences

Sample calculator display Rounded

Approximation

3.487414444 3.4874 COS

ENTER

TAN

Notes:

• If you look back at Examples 1-4, 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.

Ex. 5: Estimating Distance

• Escalators. The escalator

at the Wilshire/Vermont

Metro Rail Station in Los

Angeles rises 76 feet at a

30° angle. To find the

distance d a person travels

on the escalator stairs, you

can write a trigonometric

ratio that involves the

hypotenuse and the known

leg of 76 feet.

d76 ft

30°

Now the math d76 ft

30° sin 30° =

opposite

hypotenuse

sin 30° = 76

d

d sin 30° = 76

sin 30°

76 d =

0.5

76 d =

d = 152

Write the ratio for

sine of 30°

Substitute values.

Multiply each side by d.

Divide each side by sin 30°

Substitute 0.5 for sin 30°

Simplify

A person travels 152 feet on the escalator stairs.

Assignment:

• 1. Pgs. 434-436 (2-20 Evens, 31-34, 35-43 Odds).

• 2. Complete 30-60-90 Triangle Worksheet

• 3. Complete Rationalizing Denominators

• 4. Pgs. 441-442 (1-17 Odds, 22-24)

Exit Quiz – 10 Points

20°

x

6 cm

y

Solve for x and y. PLEASE

SHOW ALL WORK.