Ratios in Right Triangles Expectations: 1) G1.3.1: Define and use sine, cosine and tangent ratios to...

Ratios in Right Triangles

Expectations:1)G1.3.1: Define and use sine, cosine and tangent ratios to solve problems using trigonometric ratios in right triangles.2)Determine the exact values of sine, cosine and tangent for various angle measures.

04/21/23 8-3: Ratios in Right Triangles

Opposite Legs

From an acute angle in a right triangle, the leg opposite is the leg that lies in the interior of the angle (except the endpoints of the side).

C

B

A

BC is the leg opposite A

04/21/23 8-3: Ratios in Right Triangles

Opposite Legs

From an acute angle in a right triangle, the leg opposite is the leg that lies in the interior of the angle (except the endpoints of the side).

C

B

A

AC is the leg opposite ∠B

04/21/23 8-3: Ratios in Right Triangles

Adjacent Legs

The leg adjacent to an acute angle of a right triangle is the leg that forms a side of the acute angle.

C

B

A

AC is the leg adjacent ∠A

04/21/23 8-3: Ratios in Right Triangles

Adjacent Legs

The leg adjacent to an acute angle of a right triangle is the leg that forms a side of the acute angle.

C

B

A

BC is the leg adjacent ∠B

04/21/23 8-3: Ratios in Right Triangles

Sine Ratio

The sine ratio of an acute angle of a right triangle compares the length of the leg opposite the angle to the length of the hypotenuse.

Sine is abbreviated sin, but it is still read as “sine”.

04/21/23 8-3: Ratios in Right Triangles

Sine Ratio

C

B

A

sin θ = leg opposite

hypotenuse

04/21/23 8-3: Ratios in Right Triangles

Sine Ratio

C

B

A

sin A = BC

ABsin B =

ACAB

04/21/23 8-3: Ratios in Right Triangles

Cosine Ratio

The cosine ratio of an acute angle in a right triangle compares the length of the leg adjacent the acute angle to the length of the hypotenuse.

Cosine is abbreviated “cos” but is still read as “cosine.”

04/21/23 8-3: Ratios in Right Triangles

Cosine Ratio

C

B

A

cos θ = leg adjacent

hypotenuse

04/21/23 8-3: Ratios in Right Triangles

Cosine Ratio

C

B

A

cos A = AC

ABcos B =

BCAB

04/21/23 8-3: Ratios in Right Triangles

A

6

C

B

8

10

Give the sin and cos ratios for ∠A and ∠B.

04/21/23 8-3: Ratios in Right Triangles

For the right triangle shown below, what is the sin C?

a. a/b

b. a/c

c. b/a

d. c/b

e. c/a

A C

B

ab

c

Solve for x in the triangle below.

35°

24

x

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Solve for x in the triangle below.

75°x

18

04/21/23 8-3: Ratios in Right Triangles

If AC = 10 in the figure below, determine BD.

04/21/23 Trig Basics

45°

30°

a.10 2

b.5 3

c.10 3

d.5 2

e.5

Tangent Ratio

The tangent ratio of an acute angle of a right triangle compares the length of the leg opposite the acute angle to the length of the leg adjacent the acute angle.

Tangent is abbreviated “tan” but is still read as “tangent.”

04/21/23 8-3: Ratios in Right Triangles

Tangent Ratio

C

B

A

tan θ = leg opposite

leg adjacent

04/21/23 8-3: Ratios in Right Triangles

Tangent Ratio

C

B

A

tan A = BC

ACtan B =

ACBC

04/21/23 8-3: Ratios in Right Triangles

Tangent Ratio

Solve for x in the triangle below.

15

x

65°

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Solve for x and y below.

22°12

x

y

04/21/23 8-3: Ratios in Right Triangles

To guard against a fall, a ladder should form no more than a 75° angle with the ground. What is the maximum height that a 10 foot ladder can safely reach?

04/21/23 8-3: Ratios in Right Triangles

A kite is flying at the end of a 240-foot string which makes a angle with the horizon. If the hand of the person flying the kit is 3 feet above the ground, how far above the ground is the kite?

04/21/23 Trig Basics

a. 120 3 3 ftb. 120 3 3 ftc. 240 3 3 ftd. 240 3 3 fte.123 ft

Arc functions

If you know the value of a trig function, you can work backwards to determine the measure of the angle.

For example, say we know the cos A = .5, then we can use the cos-1 (arc cosine or inverse of cosine) function to determine that m∠A = 60°.

04/21/23 8-3: Ratios in Right Triangles

To calculate angles from cos:

Use the 2nd (shift or inverse) key before the cos key.

Ex: cos A = .8894Type .8894 2nd cos .This returns 27.20, so m∠ A = 27.2°You may need to type 2nd cos .8894

=04/21/23 8-3: Ratios in Right Triangles

To calculate angles from sin:

Use the 2nd (shift or inverse) key before the sin key.

Ex: sin A = .6Type .6 2nd sin .This returns 36.87, so m∠A = 36.87°You may need to type 2nd sin .6 =

04/21/23 8-3: Ratios in Right Triangles

To calculate angles from tan:

Use the 2nd (shift or inverse) key before the tan key.

Ex: tan A = .2341Type .2341 2nd tan .This returns 13.17, so m∠ A = 13.17°You may need to type 2nd tan .2341

=

04/21/23 8-3: Ratios in Right Triangles

A patient is being treated with radiotherapy for a tumor that is behind a vital organ. In order to prevent damage to the organ, the doctor must angle the rays to the tumor. If the tumor is 6.3 cm below the skin and the rays enter the body 9.8 cm to the right of the tumor, find the angle at which the rays should enter the body to hit the tumor.

04/21/23 8-3: Ratios in Right Triangles

The hypotenuse of the right triangle shown below is 22 feet long. The cosine of angle L is ¾. How many feet long is the segment LM?

A. 18.4B. 16.5C. 11.0D. 6.7E. 4.7

04/21/23 8-3: Ratios in Right Triangles

L

M N

22

Assignment

Pages 416 – 419, # 17-49 (odds), 50, 51, 53 – 65 (odds)

04/21/23 8-3: Ratios in Right Triangles