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35° 55°

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Yes, slope is more than 1 Isaiah, 19° has slope less than 1 Less than 9

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11° x

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5.4 What About Other Right Triangles? Pg. 15 The Tangent Ratio

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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?

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5.21 – LEANING TOWER OF PISA For 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.

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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.

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a. With the measurements provided, what is the slope of the leaning tower? 50 5 = 10

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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°

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c. What else can you solve for in this triangle? Find as many missing sides and angles as possible. 6° = x = x = x 2 84°

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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.

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5.22 – TRIG TABLE PRACTICE Solve 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.

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5.23 – MULTIPLE METHODS a. 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

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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

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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 aopposite adjacent 72° hypotenuse

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d. Use Eddie's observation in part (c) to confirm your answer to part (a). 72° 4 aopposite adjacent hypotenuse

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5.24 – USING A SCIENTIFIC CALCULATOR Examine 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. opposite adjacent hypotenuse

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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 hypotenuse

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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 Use the "tan" button on you calculator to verify that you get ≈ 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.

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

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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

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b. Can you use special triangle ratios to solve? Why or why not? No, angle isn’t special

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c. Can you use the Trig Table to solve? Why or why not? No, not listed

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opposite adjacent hypotenuse 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.

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opposite adjacent hypotenuse

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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. 1 opposite adjacent hypotenuse

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

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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

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5.27 – HOW TO SOLVE TANGENTS WITH CALCULATORS For 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.

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O A tan 38° = x8x8 x = 8 tan 38° x = 6.25 opposite adjacent tan θ°= H

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O A tan 73° = 9x9x x = 9/tan 73° x = 2.75 opposite adjacent tan θ° = H

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O A tan 45° = x 14 x = 14 tan 45° x = 14 opposite adjacent tan θ° = H

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O A H tan 39° = 52 x x = 52/tan 39° x = opposite adjacent tan θ° =

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O A H tan 23° = x 53 opposite adjacent tan θ° = x = 53 tan 23° x = 22.5

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O A H tan 44° = 21 x x = 21/tan 44° x = opposite adjacent tan θ° =

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Right Triangles Project Pythagorean Theorem: Given 2 sides 45º– 45º– 90º 30º– 60º– 90º Sine – S sin -1, cos -1, tan -1 Your Name Block# Cosine – C Tangent – T OHOH AHAH OAOA Clinometer Measures Area of Regular Polygons

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Tangent – T O A tan θ = OAOA OAOA

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