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Right Triangle Trigonometry

bat1.jpg

Measuring without Measuring

Did you know we can find out how big something is without even measuring it?

 

Shown to the left is Chicago's Bat Column, a sculpture by Claes Oldenburg.
How tall do you think it is?

 

  1. bat3.gifBrainstorm and come up with at least two possible ways to determine the height of the sculpture. Choose one method and write a detailed plan. (For example, you could use an extension ladder on a firetruck to climb to the top and drop a measured cord to the ground. This would be a direct measurement procedure.)

  2. An indirect way to measure the height of Bat Column would be to use a right triangle and a trigonometric ratio.
    1. In the situation drawn to the right, what lengths and angles could you determine easily by direct measurement (and without using high-powered equipment)?
    2. Which trigonometric ratios of involve side BC? Of these, which also involve a length that you could reasonably measure?
    3. Which of the trigonometric ratios of involve side BC and a measurable length? If you know the size of, how can you find the measure of ?

  3. Krista and D'wan decided to find the height of Bat Column themselves. First Krista chose a spot to be point A, 20 meters from the sculpture (point C). D'wan used a clinometer, like the one shown at the right, to estimate the measure of (the angle of elevation from the horizontal to the top of the bat). He measured to be 55°. What is the measure of ?

    Krista and D'wan proceeded to find the height of the bat independently as shown below

    D'wan

     

    boy_thinking.jpg

    dwan_bubble.gif

    krsita_bubble.gif

    girl_thinking.gif

     

    Krista
    1. Analyze D'wan's thinking. Why did he multiply by 20?
    2. Analyze Krista's thinking. Why did she multiply by BC? Why did she divide by 0.7?
    3. Are the answers correct? Explain your response.
    4. How could you use Krista's and D'wan's work to help estimate the height of Bat Column?

  4. Kim said he could find the length AB (the line of sight distance) by solving .
    kim_bubble.gif
    1. Analyze and explain Kim's thinking shown above.
    2. Is Kim correct?
    3. What is another way Kim could have found AB using trigonometric ratios?
    4. Could you find AB without using trigonometric ratios? Explain your reasoning.

  5. Let's look at a another situation where trigonometric ratios can be used to find measurements.

    Stair systems must be built in new homes according to building codes.

    If the angle of elevation (stair angle) is to be 34° and the length of each tread (the "run") is to be 10", then how big should each riser (the "rise") be for each step?

    Solution:

    So the rise for each step should be 6.745 inches or roughly 6 ¾ ".


    How much should the "rise" of a stair be if the angle of inclination is 35° and the stair treads are each 11 inches? (Show your work.)

  6. We can also can use trigonometry to find angles. Consider the following example.

    If Bob the Builder wanted to make a stair system with 10.5" treads and 7.5" risers, what angle of elevation would that be?

    Solution:

    Setting up a picture is always the best first step!

    Now the question is, what angle would give a tangent value of .7142857...?
    Either you could guess and check with your calculator to find such an angle or you can use a powerful tool that's on your calculator.

    Try to find the tan-1 or atan button on your calculator. You likely will need to use the "Shift" or "2nd" or "Inv" button first to get the tan-1 button.

    To "undo" the tangent of x, we use the "inverse tangent" button, tan-1.

    So to solve tan x° = .7142857... we have

    x = tan-1(.7142857...)

    x = 35.5376...

    You can check that tan (35.5376...) really is .7142857...
    Not many people would have been able to guess that one.

    Therefore the angle of elevation will be about 35½ °.


    If Bob the Builder wanted to make a stair system with 11" treads and 8" risers, what angle of elevation would that be? (Show your work.)

  7. Terri is flying a kite and has let out 500 feet of string. Her end of the string is 3 feet off the ground.
    1. If has a measure of 40°, what ratio of sides would sin 40° describe? (Hint: soh cah toa)
    2. Since she let out 500 feet of string, we know KI is 500. Set up the trig ratio equation for sin 40°.
    3. Solve your trig ratio equation to find how high is the kite is.
    4. As the wind picks up, Terri is able to fly the kite at a 56° angle with the horizontal. Approximately how high is the kite?
    5. What is the highest Terri could fly the kite on 500 feet of string? What would be the measure of then?
    6. Experiment with your calculator to estimate the measure of needed to fly the kite at a height of 425 feet.

  8. Set up trig equations for each of the following situations and solve for the unknown variable.


  9. Check your answers from the last problem by looking here.
    How well do you think you understand how to find missing lengths with trigonometry?

  10. Create your own problem which involves trigonometry for some situation. Provide a sample solution as well.

 

 

 

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