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?. 60 º. 5. Recall: How do we find “?”. ?. 45 º. 8. ?. 65 º. 5. What about this one?. ?. ?. ?. 60 º. 60 º. 60 º. 5. 11. 7. What is the ratio of long leg to short leg?. ?. ?. ?. 65 º. 65 º. 65 º. 5. 12. 123. These triangles are all similar (AA~).

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60

?

60º

5

Recall: How do we find “?”

?

45º

8


60

?

65º

5

What about this one?


60

?

?

?

60º

60º

60º

5

11

7

What is the ratio of long leg to short leg?


60

?

?

?

65º

65º

65º

5

12

123

These triangles are all similar (AA~).

What is the relationship of their ratios of long leg to short leg?

The ratios are all the same.


Right triangle trigonometry

Right Triangle Trigonometry

Sections 9.1 and 9.2


What is trigonometry

Angle and Side Problem

What is Trigonometry?

Side Problem

Pythagorean Theorem

Angle Problem

Triangle Sum Theorem


Tangent ratio

B

B

Opposite

Adjacent

Leg

Leg

A

A

C

C

Opposite

Adjacent

Leg

Leg

Tangent Ratio

Trig ratios are always with respect to a specific angle.


Labeling in a right triangle

Labeling in a right triangle

B

c

a

A

C

b


60

opposite

adjacent

BC

AC

20

21

tan A

=

=

=

opposite

adjacent

AC

BC

21

20

tan B

=

=

=

Write the tangent ratios for A and B.


Calculator trig functions

Calculator Trig Functions

B

37

°

A

C

If you must round, use at least 3 decimal places.

Make sure the calculator  is set to “degrees”


60

Use the tangent ratio.

height

125

tan 32° =

height = 125 (tan 32°)

Solve for height.

Use a calculator.

125 32 78.108669

To measure the height of a tree, Alma walked 125 ft from the tree and measured a 32° angle from the ground to the top of the tree. Estimate the height of the tree.

The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree.

The tree is about 78 ft tall.


Sine ratio

B

Hypotenuse

A

C

Opposite

Leg

Sine Ratio

B

Hypotenuse

Opposite

Leg

A

C


Cosine ratio

B

Hypotenuse

Adjacent

Leg

A

C

Cosine Ratio

B

Hypotenuse

A

C

Adjacent

Leg


60

opposite

hypotenuse

12

20

3

5

sin T =

=

=

adjacent

hypotenuse

16

20

4

5

cos T =

=

=

opposite

hypotenuse

16

20

4

5

sin G =

=

=

adjacent

hypotenuse

12

20

3

5

cos G =

=

=

Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms.


Calculator trig functions1

Calculator Trig Functions

B

37

°

A

C

Make sure the calculator  is set to “degrees”


60

Use the cosine ratio.

height = 20 • cos 35°

Solve for height.

Use a calculator.

height

20

cos 35° =

20 35 16.383041

A 20-ft. wire supporting a flagpole forms a 35˚ angle with the flagpole. To the nearest foot, how high is the flagpole?

The flagpole, wire, and ground form a right triangle with the wire as the hypotenuse.

Because you know an angle and the measures of its adjacent side and the hypotenuse, you can use the cosine ratio to find the height of the flagpole.

The flagpole is about 16 ft tall.


Soh cah toa

SOH-CAH-TOA

SOH

CAH

TOA

SOH-CAH-TOA


Inverse trig functions

Inverse Trig Functions

B

If the Sin of an angle is 0.8191, what is the measure of the angle?

x

°

A

C


Regular vs inverse

Regular vs. Inverse


60

A right triangle has a leg 1.5 units long and hypotenuse 4.0 units long. Find the measures of its acute angles to the nearest degree.

Draw a diagram using the information given.

Use the inverse of the cosine function to find m A.

1.5

4.0

cos A =

=

0.375

Use the cosine ratio.

Use a calculator.

Use the inverse of the cosine.

m A = cos–1(0.375)

Round to the nearest degree.

0.375 67.975687

m A 68


60

(continued)

To find m B, use the fact that the acute angles of a right triangle are complementary.

m A + m B = 90

Definition of complementary angles

68 + m B 90

Substitute.

mB 22

The acute angles, rounded to the nearest degree, measure 68 and 22.


60

Find m R to the nearest degree.

47

41

tan R =

Find the tangent ratio.

47

41

Use the inverse of the tangent.

47

41

Use a calculator.

48.900494

m R tan–1

So m R 49.


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