Download Trigonometry in general triangles

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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
1. In commercials, the Straight Shot Screamer is advertised as “120 feet of pure
excitement!” Use what you know about right triangle trigonometry to determine the
unknown side lengths and angle measure.
77.786
59.588
65.270
24.624
2. Using the completed triangle from question 1, write two different sine ratios that involve
the length 50 ft. Isolate the 50 in each equation and write a relationship between the
two sine values and sides of the triangle.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
3. Fill in the blanks to develop the proof of the sine-to-side relationship.
c sinA = a sinC
h
h
sinA = 1 ; sinC = 1
c
a
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
c sinA = h1; a sinC = h1
sinA sinC
=
a
c
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
4. Fill in the table with the trig ratios for some special angles and their supplements. For
each angle, think about what the sides of the reference triangle would be and calculate
each trig ratio.
a°
sin a°
cos a°
tan a°
30°
45°
60°
120°
135°
150°
5. Fill in the blanks to summarize your findings about the sine value of supplementary
angles.
supplementary
180°
90°
complementary
are always equal
always add up to 1
The degree measures of two
angles add up to
.
The degree measures of two
angles add up to
.
The sine values of two
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
angles
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.
Student:
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
6. In the triangle above, what is the relationship between ∠ABC and ∠ABD? What does this
tell you about the sine value of the two angles?
7. Use ∆ABD to help you write a ratio for sin ∠ABC.
8. You have two sine ratios involving h2: sin B =
and sin C =
. Use the two ratios to
write the sine-to-side relationship from earlier. Start by isolating h2 in each ratio.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
9. Write the Law of Sines.
10. Use the Law of Sines to find the base of the Straight Shot Screams, from the foot of the
steps to the end of the slide. (Hint: use the 40° ratio since it does not contain any
approximate values.)
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
11. R EINFORCE Use the Law of Sines to solve for the missing side lengths or angle measures
in the examples below.
a.
x=
y=
z=
x=
y=
z=
b.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
12. Sketch the different kinds of triangles formed by the Dunkin’ Swing ride.
13. When two sides and one nonincluded angle are given, describe the types of triangles that
might result.
14. In each of the triangles below, you are given two side lengths and the measure of the
nonincluded angle. Solve each of the triangles for the missing angle measure. Explain
your solutions.
a.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
b.
c.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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Date:
Student:
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Date:
Trigonometry in general triangles
Student Activity Sheet 2; use with Exploring “Law of Sines”
15. R EINFORCE
a. In ΔABC, m∠A = 36º, a = 7, and b = 10. Determine whether ∠B exists. If so, find all
possible measures of ∠B.
b. Sketch all possible triangles with these measures.
16. R EINFORCE From fire towers Q and R, located 18 miles apart, a fire is sighted at point
F. If m∠FRQ = 70º and m∠FQR = 48º, find the distance (to the nearest mile) from point
F to the closest fire tower. Sketch a diagram as part of your solution.
Copyright 2013 Agile Mind, Inc. ®
Content copyright 2013 Charles A. Dana
Center, The University of Texas at Austin
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