Download HSCE: G1 - Math Companion Documents

HSCE: G1.3.1 Define the sine, cosine, and tangent of acute angles in a right
triangle as ratios of sides; solve problems about angles, side
lengths, or areas using trigonometric ratios in right triangles.
Clarification statements:
The sine of an angle is the ratio of the length of the side opposite to angle to
the length of the hypotenuse of the right triangle (sine = opp / hyp)
The cosine of angle is the ratio of the length of the side adjacent to the angle
to the length of the hypotenuse of the right triangle (cos = adj / hyp).
The tangent of an angle is the ratio of the length of the side opposite the
angle to the length of the side adjacent to the angle (not the hypotenuse)
(tan = opp / adj).
Mnemonic devices:
Oscar Had A Heavy Old Anvil (Opposite over Hypotenuse; Adjacent over
Hypotenuse; Opposite over Adjacent for sine, cosine, and tangent,
respectively)
SOH-COA-TOA (pronounced so-ca-toa)
Example 1:
A.
7
14
B.
7
25
C.
14
7
D.
24
25
E.
25
24
Note: In above triangle, students should also be able to find cosine and
tangent of angle A. Students should also be able to find the length of a side
given an angle and a side.
Example 2:
Draw right triangle ABC, where angle A is 30 degrees and angle C is the right
angle (90 degrees). The length of side BC is 5 units. Find the length of side
AB.
Example 3:
Project to Discover Trig Ratios
Adapted from Contemporary Mathematics in Context, course 2B
a. Each group member should draw a segment AS (each a different
length – lengths of 15 cm, 20 cm, and 25 cm work well). Using your
segment AC as a side, draw triangle ABC with angle A measuring 35
degrees and a right angle at C. It is important to draw your triangle
very carefully. What is the measure of the other angle (angle B)?
b. Are the triangles your group members drew similar? Explain.
c. Measure the 3 sides of your triangle. Does a2 + b2 = c2? (Connection
to prior knowledge regarding Pythagorean Theorem)
d. Choose the smallest triangle drawn in Part a. Determine the
approximate scale factors relating this triangle to the others drawn by
group members.
For a right triangle ABC, it is standard procedure to label the right
angle with the capital letter C and to label the sides opposite the three
angles lower case a, b, and c as shown. Complete a labeling if your
triangle in this way. The hypotenuse is always the side opposite the
right angle
e. Make a table like the one below. Each group
member should choose a unit of measure. Then
carefully measure and calculate the
indicated ratios for the right triangle ABC
that you drew. Express the ratios to
the nearest 0.01. Investigate
patterns in the three ratios
for your group’s triangles.
Right Triangle Side Ratios (35-55-90)
Ratio Student 1 Student 2 Student 3 Student 4
a/b
b/c
a/b
f. Compare the ratios from your group with those of other groups.
g. Make a conjecture about the three ratios in the table for any right
triangle ABC with a 35 degree angle A.
h. How could the three ratios be described in terms of the hypotenuse
and the sides opposite and adjacent to angle A? In terms of the
hypotenuse and the sides opposite and adjacent to angle B?
i. Make a conjecture about the three ratios in the table for any right
triangle with a 35 degree angle.
j. Make a conjecture about the three ratios for any right triangle with a
55 degree angle.
k. Repeat the activity above with a 40-50-90 degree triangle.
HSCE: G1.3.2 Know and use the Law of Sines and the Law of Cosines and
use them to solve problems; find the area of a triangle with
sides a and b and included angleˆ using the formula Area =
(1/2) a b sinˆ.
Clarification statements:
The Law of Sines and the Law of Cosines are usually used in non-right
triangles. (The laws hold for right triangles, but they are not necessary,
because right-triangle definitions of sine and cosine are more useful in a
right triangle.)
Law of Sines
sin A sin B sin C


a
b
c
This means that the ratio of the sine of angle A to the length of the side
opposite angle A is the same as the ratio of the sine of angle B to the length
of the side opposite angle B, and is the same as the ratio of the sine of
angle C to the length of the side opposite angle C, for all triangles.
Law of Cosines
c2 = a2 + b2 -2ab cos C
b2 = a2 + c2 – 2ac cos B
a2 = b2 + c2 - 2bc cos A
In the formulas above, a, b, and c refer to the
the sides opposite angles A, B, and C, respectively.
lengths of
In deciding whether to use the Law of Sines or the Law of Cosines, it is
helpful to look at what information you have. For example, if you know the
lengths of the three sides of the triangle (SSS), or if you know the lengths of
two of the sides and the measure of the angle in between (SAS), the Law of
Cosines is useful. For other triangles, use the Law of Sines.
Example 1:
Law of Sines and Law of Cosines:
http://illuminations.nctm.org/LessonDetail.aspx?id=U177
In this unit, students learn the Law of Sines and the Law of Cosines and
determine when each can be used to find a side length or angle of a triangle.
Lesson 1 - Law of Sines
In this lesson, students will use right triangle trigonometry to develop the law
of sines.
Lesson 2 - Law of Cosines
In this lesson, students use right triangle trigonometry and the Pythagorean
theorem to develop the law of cosines.
Example 2:
For triangle ABC, with sides of length a, b, and c, the Area = ½ ab sin C,
where C is the angle between sides a and b. To develop this formula, begin
with the general formula for area of a triangle: A = ½ bh (Area = ½ * base
* height). Using right-triangle definitions, the sine of angle C is ratio of the
length of the side opposite the angle to the length of the side adjacent to the
angle. In the diagram below, the base of the triangle is a, and sin C =
h
.
b
Solving for h, h = b sin C. Therefore, Area = ½ ah = ½ ab sin C.
Web Links:
The above image, along with a complete explanation for the formula, can be
found at http://regentsprep.org/Regents/mathb/5E1/areatriglesson.htm
For more background on this formula, see Dave’s Short Trig Course at
http://www.clarku.edu/~djoyce/trig/area.html
Another proof of the formula, along with step-by-step solutions to an
example problem can be found at the Name Project at
http://www.acts.tinet.ie/areaofatriangle_673.html
HSCE: G1.3.3 Determine the exact values of sine, cosine, and tangent for 0°,
30°, 45°, 60°, and their integer multiples, and apply in various
contexts.
Example 1:
Web Link http://www.karlscalculus.org/calc7_0.html
This site explores the unit circle in context of a scenario in which students
raise the lost plateau of Atlantis in order to create new land on which to
support a growing population.
Example 2:
Students Develop Sine, Cosine, and Tangent
Prior Knowledge: Prior to completing the following activity, students need to
know G1.2.4 (relationships among side lengths and the angles of 30-60-90
triangles and 45-45-90 triangles) and G1.3.1 (definition of sine, cosine, and
tangent).
1. Draw right triangle ABC, with angle A = 30 degrees, angle B = 60
degrees, and angle C = 90 degrees. Label side a (opposite angle A)
with a length of a. Using the ratios of the sides in a 30-60-90
triangle, identify the lengths of the other sides in terms of a.
(Answers: c = 2a; b = a 3 )
2. Use the right-triangle definitions of sine and cosine (sine = opposite /
hypotenuse; cosine = adjacent / hypotenuse) to find the sine and
cosine for 30 degrees and 60 degrees.
3. Repeat the above process for a 45-45-90 triangle, labeling the lengths
of the legs as a.
a 1

2a 2
a 3
cos 30 

2a
a 3
sin 60 

2a
a 1
cos 60 

2a 2
sin 30 
sin 45 
cos 45 
a
a 2

1
1
2
2



2
2
2
2

1
1
2
2



2
2
2
2
a
a 2
a
tan 45   1
a
3
2
3
2