6/24/2010 ©evergreen public schools 2010 1 lesson title teacher notes supplies: scientific...

6/24/2010

©Evergreen Public Schools 2010

1

Lesson TitleTeacher Notes

Supplies: scientific calculators for all kids

Notes: The goal for this lesson is to help students build on the Exploring Triangles Lab and the Special Right Triangles work with ratios of side lengths. The push is to help students understand that the trig ratios are ratios that compare the side lengths, a concept that many students do not fully understand.

Vocabulary: opposite, adjacent, sine, cosine, tangent


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I can find the side lengths of any right triangle using trigonometric ratios.

What are some of the ratios about triangles that we’ve already

considered in this unit?


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LaunchLaunchLaunchLaunchFind all of the missing angles and side lengths of the following right triangle.How are you able to determine them all?

30

8 cm


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26-64-90 Triangles26-64-90 Triangles

On your student notes page, sketch a 26-64-90 triangle with a short leg of 20 cm.

On a 26-64-90 triangle, which angle is opposite the shortest side?Which angle is opposite the longest side (the hypotenuse)?

20 cm


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Using the ratios of the side lengths that you found during the Exploring Triangles Lab, find the lengths of the other 2 sides.

Check your answers with your partner. How did you find them? Did you and your partner find them the same way?

20 cm

26

64


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Sketch a 26-64-90 triangle with a hypotenuse of 55 cm. Include all of the angle measures and the other two side lengths.


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37-53-90 Triangle37-53-90 Triangle

Sketch a 37-53-90 triangle with a long leg of 32 cm.How do you know which angle is opposite the side length of 32 cm?Which angle is adjacent to the side length of 32 cm?

32 cm


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37-53-90 Triangle37-53-90 Triangle

Using the ratios you found for 37-53-90 triangles in the Exploring Triangles Lab, find the other two side lengths and label the triangles’ side lengths and angle measures.

32 cm


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Similar TrianglesSimilar Triangles

Triangle 13, 17 and 18 in Exploring Triangles were similar triangles because their angles were all congruent …


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and because their sides were proportional.


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Mathematicians used those characteristics of similar triangles to make tables of information based on angle measurements. No matter how long the sides are, if the angles are the same, the ratios of the sides will be, as well.


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Right triangle trigonometry developed from the very unique relationships among similar triangles.


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Trigonometric RatiosTrigonometric Ratios

A trigonometric ratio tells you the ratio of two sides of a right triangle in connection to one of the two non-right angles.

The sine of 53 is 0.8 – that means the ratio of the side opposite the 53 angle hypotenuse

= 0.853

opposite

hypotenuse


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sin 53 = side opposite the 53 angle

hypotenuse

How could you use this ratio and the length of the hypotenuse to find the length of the leg opposite the 53º angle?

= 0.853

opposite

40 cm


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The cosine of 53 is the ratio of theside adjacent to the 53 angle hypotenuse

What is the value of the cosine of 53? How do you know?

24 cm40 cm53º


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The tangent of 53 is the ratio of the side opposite the 53 angle side adjacent to the 53 angle

The tangent of 53 is 1.3.

Where do you see the tangent of 53 ?

24 cm

32 cm

53º


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On your student notes page, write the following definitions. “A” represents one of the non-right angles in a right triangle.Sine: sin A = side opposite A s = o

hypotenuse h

Cosine: cos A = side adjacent to A c = a hypotenuse h

Tangent: tan A = side opposite A t = o side adjacent to A a


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Use the 37-53-90 triangle to find the sine, cosine and tangent of the 37º angle.

sin 37º =

cos 37º =

tan 37º =

y cm

x cm

40 cm

37º


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Team PracticeTeam Practice

Write a trig ratios that could be used to approximate the value of x length to the nearest tenth. Then use a calculator to determine the value. a. b.

42º

8

x

21º

5

x


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DebriefDebrief

With your elbow partner,• One person describe what

the sine of an angle is.

• The other describe how the sine is similar to and different from the cosine.

• Be prepared to share what you discussed with the class.


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5

3

12

4

Did you hit the target? I can find the side lengths

of any right triangle using trigonometric ratios.

Rate your understanding of the target from 1 to 5.

(Not to worry – we’ll do more work with trig ratios!)


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PracticePractice

Practice Sheet Unit 2, 4-1.


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Ticket Out

What is one question that you have about trigonometric ratios?

6/24/2010 ©evergreen public schools 2010 1 lesson title teacher notes supplies: scientific...

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