x 19

11y 175

35°55°

1

5

10

xx 50

5

2

10

xx 4

AB

Yes, slope is more than 1

Isaiah, 19° has slope less than 1

Less than 9

11°120 + 50

x

1

5

170

xx 850

5.4

What About Other Right Triangles?

Pg. 15The Tangent Ratio

5.4 – What About Other Right Triangles?The Tangent Ratio

In this chapter you started a Trig table of angles and their related slope ratios. Unfortunately, you only have the information for a few angles. How can you quickly find the ratios for other angles when a computer is not available or when an angle is not on your Trig table? Do you have to draw each angle to get its slope ratio? Is there another way?

5.21 – LEANING TOWER OF PISAFor centuries, people have marveled at the Leaning Tower of Pisa due to its slant and beauty. Ever since construction of the tower started in the 1100's, the tower has slowly tilted south and has increasingly been at risk of falling over. It is feared that if the angle of slant ever falls below 83°, the tower will collapse.

Engineers closely monitor the angle at which the tower leans. With careful measuring, they know that the point labeled A in the diagram at right is now 50 meters off the ground. Also, they determined that when a weight is dropped from point A, it lands five meters from the base of the tower, as shown in the diagram.

a. With the measurements provided, what is the slope of the leaning tower?

505

= 10

b. Use your Trig Table to determine the angle at which the Leaning Tower of Pisa slants. Is it in immediate danger of collapse?

84°Not yet!

84°

c. What else can you solve for in this triangle? Find as many missing sides and angles as possible. 6°

52 + 502 = x2

25 + 2500 = x2

2525 = x2

5 101 x84°

The construction started in the year 1174 by Bonanno Pisano. When the tower had reached its third story the work ceased because it had started sinking into the ground. The tower remained thus for 90 years. It was completed by Giovanni di Simone, Tommano Simone (son of Andreo Pisano), crowned the tower with the belfry at half of 14th century.

It started to lean due to the sinking of the ground right from the time of its construction. Unfortunately, even today the great mass continues to sink very slowly. It is a question of about 1 mm. every year.

5.22 – TRIG TABLE PRACTICESolve for the variables in the triangles below. It may be helpful to first orient the triangle (by rotating your paper or by using tracing paper) so that the triangle resembles a slope triangle. Use your Trig Table for reference.

5

1

12

w

2.4w

5 12w

11

2

5

8

z

3.2z

5 16z

68

1

3

15

x

5x

3 15x

72

5.23 – MULTIPLE METHODSa. Tanya, Mary, Eddie, and Amy are looking at the triangle below and trying to find the missing side length. Tanya declares, "Hey! We can rotate the triangle so that 18° looks like a slope angle, and then ∆y = 4." Use her method to solve for a.

18°4

a

1

3

4

a

12a

b. Mary says, "I see it differently. I can tell ∆y = 4 without turning the triangle." How can she tell? Explain one way she could know.

opposite

adjacent

hypotenuse

c. Eddie replies, "What if we use 72° as our slope angle? Then ∆x = 4." What is he talking about? Discuss with your team and explain by redrawing the picture and using words.

72°

4

a opposite

adjacent

72°hypotenuse

d. Use Eddie's observation in part (c) to confirm your answer to part (a).

3

1

4

a

12a72°

4

a opposite

adjacent

hypotenuse

5.24 – USING A SCIENTIFIC CALCULATORExamine the triangle at right.a. According to the triangle at right, what is the slope ratio for 32°? Explain how you decided to set up the ratio. Write the ratio in both fraction and decimal form. Round to 4 decimal places.

oppositeadjacent

hypotenuse

5

80.625

b. What is the slope ratio for the 58° angle? Write the ratio in both fraction and decimal form. How do you know?

opposite adjacent

8

51.6 hypotenuse

c. Scientific calculators have a button that will give the slope ratio when the slope angle is entered. In part (a), you calculated the slope ratio for 32° as 0.625. Use the "tan" button on you calculator to verify that you get ≈ 0.625 when you enter 32°. On some calculators you type in "tan (32)" and in others you type the degree in first with "32 tan". Be ready to help your teammates find the button on their calculator. Make sure you are in DEGREE mode.

http://i2.expansys.com/img/b/106385/texas-inst-ti-30xiis-scientific-calculator.jpg

tan 32° = _________________ d.Does that button give you ≈ 1.600 when you enter tan 58°? Be ready to help your teammates find the button on their calculator.

tan 58° = _________________

0.625

1.6

5.25 – WHAT CAN WE USE? Examine the triangle at right.  a. Can you solve for x using the Pythagorean theorem? Why or why not?

No, only know one side

b. Can you use special triangle ratios to solve? Why or why not?

No, angle isn’t special

c. Can you use the Trig Table to solve? Why or why not?

No, not listed

op

po

site

adjacent

hypote

nuse

d. Merisa decided to solve the triangle using the relationship she just discovered using tangents. She said, "I can label my triangle to show what side is opposite the angle and what side is adjacent." How do you know what side is opposite and what side is adjacent. Discuss this with your team and label the triangle.

tan

opposite side

adjacent side op

po

site

adjacent

hypote

nuse

tan33 12

x

f. Use the equation you set up in part (e) to solve for the variable. DO NOT ROUND for the tangent slope ratio. You should use the exact value from your calculator. Your final answer can be rounded to two decimal places. Be sure to help your group with their calculators.

tan33 12

x

1

x 12 tan33 7.79x

op

po

site

adjacent

hypote

nuse

5.26 – MULTIPLY OR DIVIDE?Examine the proportions below. a. Solve the proportions below. Do not round until your final answer.

4 tan 25x

1.87x 4 tan 25x 8.58x

tan 25 4x

6 tan55x 4.20x

tan55 6x 6 tan55x

8.57x

b. When did you multiply the number with the tangent angle? Why?

c. When did you divide the number by the tangent angle? Why?

When the variable is on the top

When the variable is on the bottom

5.27 – HOW TO SOLVE TANGENTS WITH CALCULATORSFor each triangle below, label the opposite and adjacent side based on the reference angle you chose. Write an equation like you did in Problem 4.19(e) and solve for x. Be sure to show all your steps. Round your final answer to two decimal places.

O

A

tan 38° = x8

x = 8 tan 38° x = 6.25

oppositeadjacent

tan θ°=

H

OA

tan 73° = 9x

x = 9/tan 73° x = 2.75

oppositeadjacent

tan θ° =

H

O

A tan 45° = x14

x = 14 tan 45° x = 14

oppositeadjacent

tan θ° = H

O

A

H

tan 39° = 52x

x = 52/tan 39°

x = 64.21

oppositeadjacent

tan θ° =

OA

H

tan 23° = x53

oppositeadjacent

tan θ° =

x = 53 tan 23°

x = 22.5

O

A

Htan 44° = 21

xx = 21/tan 44°

x = 21.75

oppositeadjacent

tan θ° =

Right Triangles Project

Pythagorean Theorem: Given 2 sides

45º– 45º– 90º

30º– 60º– 90º

Sine – S

sin-1, cos-1, tan-1

Your NameBlock#

Cosine – C

Tangent – T

2x x x 3 2x x x

OH

AHOA

Clinometer MeasuresArea of Regular Polygons

Tangent – T

O

A

tan θ = OA

OA