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8.3 Trigonometric Ratios-Tangent Geometry Mr. Peebles Spring 2013

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8.3 Trigonometric

Ratios-Tangent Geometry

Mr. Peebles

Spring 2013

Page 2: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Bell Ringer

5

3

Page 3: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Bell Ringer

3 • 5

5 • 5

Multiply the numerator and

denominator by 5 so the denominator

becomes a whole number.

= 3

5

a.

15

5 =

5

3

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Daily Learning Target (DLT)

Monday April 15, 2013 • “I can apply my knowledge of right triangles

to find the tangent of an acute angle.”

Page 5: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Finding Trig Ratios

• A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.

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Trigonometric Ratios-

Tangent • Let ∆ABC be a right

triangle. The

tangent of the

acute angle A is

defined as follows.

ac

bside adjacent to angle A

Side

opposite

angle A

hypotenuse

A

B

C

tan A = Side opposite A

Side adjacent to A

= a

b

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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.

Page 8: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 1: Find Tan Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small-Reduced

tanA = opposite

adjacent

Trig ratios are often expressed as decimal approximations.

Page 9: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 1: Find Tan Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small-Reduced

tanA = opposite

adjacent 8

15 ≈ 0.5333 4

7.5 ≈ 0.5333

Trig ratios are often expressed as decimal approximations.

Page 10: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 2: Find Tan Ratios

S

tanS = opposite

adjacent

adjacent

opposite

12

13 hypotenuse5

R

T S

Page 11: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 2: Find Tan Ratios

S

tanS = opposite

adjacent 5

12 ≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

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Ex. 2: Finding Trig Ratios—Find the

tangent of the indicated angle.

R

tanS = opposite

adjacent

adjacent

opposite12

13 hypotenuse5

R

T S

Page 13: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 2: Finding Trig Ratios—Find the

tangent of the indicated angle.

R

tanS = opposite

adjacent 12

5 ≈ 2.4

adjacent

opposite12

13 hypotenuse5

R

T S

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Ex. 3: Finding Trig Ratios—Find the

tangent of 45

45

opposite

adjacent

1

hypotenuse1

√2

tan 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

Page 15: 8.3 Trigonometric Ratios-Tangent · 8.3 Trigonometric Ratios-Tangent Geometry ... •A trigonometric ratio is a ratio of the ... 9.5 Trigonometric Ratios

Ex. 3: Finding Trig Ratios—Find the

tangent of 45

45

opposite

adjacent

1

hypotenuse1

√2

tan 45= 1

1 = 1

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

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2

1

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

the cosine, and the tangent of 30

30

opposite

adjacent

√3

tan 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

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2

1

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

the cosine, and the tangent of 30

30

opposite

adjacent

√3

tan 30= √3

3 ≈ 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

√3

1 =

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Ex: 5 Using a Calculator

• You can use a calculator to

approximate the tangent of 74. Make

sure that your calculator is in degree

mode. The table shows some sample

keystroke sequences accepted by

most calculators.

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Sample keystrokes Sample keystroke

sequences

Sample calculator display Rounded

Approximation

3.487414444 3.4874 74

74

TAN

TAN

ENTER

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Using Trigonometric Ratios in

Real-life

• Suppose you stand and look up at a

point in the distance. Maybe you are

looking up at the top of a tree as in

Example 6. The angle that your line

of sight makes with a line drawn

horizontally is called angle of

elevation.

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Ex. 6: Indirect Measurement • You are measuring the height

of a Sitka spruce tree in

Alaska. You stand 45 feet

from the base of the tree.

You measure the angle of

elevation from a point on the

ground to the top of the top of

the tree to be 59°. To

estimate the height of the

tree, you can write a

trigonometric ratio that

involves the height h and the

known length of 45 feet.

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The math

tan 59° = opposite

adjacent

tan 59° = h

45

Write the ratio

Substitute values

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The math

tan 59° = opposite

adjacent

tan 59° = h

45

45 tan 59° = h

45 (1.6643) ≈ h

74.8 ≈ h

Write the ratio

Substitute values

Multiply each side by 45

Use a calculator or table to find tan 59°

Simplify

The tree is about 75 feet tall.

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Assignment:

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

• 2. Complete 30-60-90 Triangle Worksheet

• 3. Complete Rationalizing Denominators

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Geometry Testing Procedures 1. Sit every other seat for the test.

2. Everything put away except something to

write with and a calculator.

3. Eyes on your own test and answer sheet.

4. NO ELECTRONICS OUT… CALCULATORS

ARE OK!

5. YOU MAY WRITE ON THE TEST.

6. WORK QUIETLY AND BE RESPECTFUL THE

ENTIRE PERIOD.

7. QUIETLY Complete 8.3 Assignment when

done.

8. DISRUPTION = CONSEQUENCE