Tangent Activity Exploration

Auteur :Bradford L Dalton, Tim Brzezinski, glassfordpeak

Thème :Cosinus, Rapports, Triangles Rectangles, Sinus, Triangles, Trigonometrie

What's Behind the Tangent of an Acute Angle?

For this activity, we will only use the TANGENT slider, the point B and adjust the angle by typing in the input box to conceptually see the meaning of the tangent ratio within this right triangle context. Spend a few minutes interacting with the applet. You can move the 2 LARGER WHITE VERTICES of the triangle anywhere you'd like and drag the triangle to different areas on the grid. The black ACUTE ANGLE slider (or input box below it) is used to change the size of angle

. Then see below for instructions.

EXPLORATORY QUESTIONS

1.)

What does the triangle look like when the sliders (tangent and acute angle) are set to the far left? Describe what you see. Now move the TANGENT slider to the far right.

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2.)

2. Reset the TANGENT slider to the far left. Adjust the acute slider to 15 degrees. Describe what you see. Now move the TANGENT slider to the far right.

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3.)

Reset the TANGENT slider to the far left. Adjust the acute angle so that it's bigger than 15 degrees but less than 45 degrees. Move the TANGENT slider to the right. Describe what you see. Explain the numerical values and words in the upper left corner. The _________ leg is ____% as long as the _______ leg.

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4.)

Reset the TANGENT slider to the far left. Set the acute angle to 45 degrees. Describe what you see.

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5.)

Reset the TANGENT slider to the far left. Set the acute angle to a value between 45 and 89 degrees...say 60 degrees. Describe what you see.

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6.)

Reset the TANGENT slider to the far left. Set the acute angle to 90 degrees. Describe what you see.

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Exploration part 2

Now lets explore what happens to the ratio of opposite side to the adjacent side as we keep the angle constant and change the side length AB. Make sure that point A of the triangle is centered at (0,0) and that the adjacent side is coincedent with the x-axis. As you slide point B along the x-axis make sure it "locks" into place at the specified lengths. We will change the side length to these lengths (1,2,3,4,5,6,7,8) and collect the data in a chart to show the changes. We will calculate the ratio of the fixed angle 50.19 degrees which we will type in the input box for acute angle.

Data table for the Tangent ratio for a fixed angle of 50.19 degrees

1.)

What happens to the ratio of the adjacent to opposite sides as the adjacent side changes but the angle measure stays constant?

The Tangent ratio is the length of the side opposite a given angle to the length of the side adjacent to that angle in a right triangle

What conclusions can you make about the Tangent ratio in any right triangle if you know the angle(s) of the right triangle.

Tangent Ratio

If you need to explore a little more to answer the following go back and explore some more. Can you conclude that if you know the ratios of the leg lengths in a right triangle then you could also find the missing angles? Explain

Tangent Activity Exploration

What's Behind the Tangent of an Acute Angle?

1.)

2.)

3.)

4.)

5.)

6.)

Exploration part 2

Data table for the Tangent ratio for a fixed angle of 50.19 degrees

1.)

The Tangent ratio is the length of the side opposite a given angle to the length of the side adjacent to that angle in a right triangle

Tangent Ratio

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