Download Module 2 Lesson 28 with Notes

GEOMETRY
MODULE 2 LESSON 28
SOLVING PROBLEMS USING SINE AND COSINE
OPENING EXERCISE
Complete the table below without looking at previous notes.
𝜽
𝟎°
𝟑𝟎°
𝟒𝟓°
𝟔𝟎°
𝟗𝟎°
Sine
0
1
2
√2
2
√3
2
1
Cosine
1
√3
2
√2
2
1
2
0
WORKBOOK

Complete Exercise 1a with a partner. You may use the information below to help you.
Let x represent the distance from school to Janneth’s house. Since sin 41 =
5.3
, then
8
300
𝑥
=
5.3
8
.
𝑥 = 452.8301887 …
For the rest of the lecture, you will need a calculator with trigonometric functions. Be sure you are in
degree mode. Together we will use calculators to find the value of sin 10°.

Complete the table in Exercise 2. Round all results to the ten-thousandth place.

What do you notice about the numbers in the row sin 𝜃 compared with the numbers in the
row cos 𝜃?
The numbers are the same but reversed in order.
Let’s find the values of a and b. Round final results to two decimal places.
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We can find the length of side a using sin 40 and cos 50.
𝑎
sin 40 =
26
26 sin 40 = 𝑎
26(0.6428) ≈ 𝑎
𝑎 ≈ 16.71
We can find the length of side b using sin 50 and cos 40.
sin 50 =
𝑏
26
26 sin 50 = 𝑏
26(0.7660) ≈ 𝑏
𝑏 ≈ 19.92

Complete Exercise 5.
A shipmate set a boat to sail exactly 27°
NE from the dock. After traveling 120
miles, the shipmate realized he had
misunderstood the instruction from the
captain; he was supposed to set sail going
directly east!
a. How many miles will the shipmate have to travel directly south before he is directly east of the
dock? Round your answer to the nearest mile.
Let S represent the distance they travel directly south.
sin 27 =
𝑆
120
𝑆 = 120 sin 27 = 54.47885997 …
He traveled approximately 54 miles south.
b. How many extra miles does the shipmate travel by going the wrong direction compared to
going directly east? Round your answer to the nearest mile.
Let E represent the distance they travel directly south.
cos 27 =
𝐸
120
𝐸 = 120 cos 27 = 106.927829 …
He traveled approximately 107 miles east
The total distance traveled by the boat is 120 + 54 = 174. They ended up 107 miles east of the dock. So they
traveled 174 – 107 = 67 extra miles.
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DISCUSSION
Johanna borrowed some tools from a friend so that she could precisely, but not exactly, measure the
corner space in her backyard to plant some vegetables. She wants to build a fence to prevent her dog
from digging up the seeds that she plants. Johanna returned the tools to her friend before making the
most important measurement: the one that would give the length of the fence!
Johanna decided that she could just use the Pythagorean theorem
to find the length of the fence she would need. Is the Pythagorean
theorem applicable in this situation? Explain.
No. The corner of her backyard is not a 90° anlge.
What can we do to help Johanna figure out the length of fence she needs? Consider the following.
The missing side is equal to 𝑥 + 𝑦.
𝑥
cos 35 =
100
𝑥 = 100 cos 35
cos 50 =
𝑦
74.875
𝑦 = 74.875 cos 50
𝑥 + 𝑦 = 100 cos 35 + 74.875 cos 50 ≈ 81.92 + 48.12872 ≈ 130.05
ON YOUR OWN
Complete Exercise 6 in your workbook.
𝑥 + 𝑦 = 4.04 cos 39 + 3.85 cos 42 ≈ 3.139669 + 2.861107 ≈ 6.000776
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SUMMARY
Solving Right Triangles

If two sides are known, then the Pythagorean theorem can be used to determine the length of the
third side.

If one side is known and the measure of one of the acute angles is known, the sine, cosine, or
tangent can be used.

If the triangle is known to be similar to another triangle where the side lengths are given, then
corresponding ratios or knowledge of the scale factor can be used to determine the unknown
length.

Direct measurement can be used.
Solving Other Triangles

You can find the length of an unknown side length of a triangle when you know two of the side
lengths and the missing side is between two acute angles. Split the triangle into two right triangles,
and find the lengths of two pieces of the missing side.
HOMEWORK
Problem Set Module 2 Lesson 28, page 217
#1, 2a and b, 3, 4, 6: Show all work in an organized and linear manner.
̅̅ can make
NOTE: For Problem 4, there are two correct answers to this problem since the segment ̅̅
𝐶𝑆
an angle of 48° above or below the horizon in four distinct locations, providing two different heights
above the ground. Choose the angle below the horizon.
DUE: Tuesday, Jan 17, 2017
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