Download Law of Sines

Trigonometry
PACKET
DATE
HOMEWORK
DATE
HOMEWORK
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HELPFUL WEBSITES:
Law of Sines:
1. (VIDEO) http://www.brightstorm.com/math/trigonometry/basic-trigonometry/the-law-of-sines/
2. (NOTES) http://www.regentsprep.org/regents/math/algtrig/ATT12/lawofsines.htm
3. (PRACTICE) http://www.regentsprep.org/regents/math/algtrig/ATT12/lawsinespractice.htm
Law of Cosines:
1.
(VIDEO) https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/law-sines-cosines/v/lawof-cosines-example
2.
(NOTES) http://www.regentsprep.org/regents/math/algtrig/ATT12/lawofcosines.htm
3.
(PRACTICE) http://www.regentsprep.org/regents/math/algtrig/ATT12/lawcosinespractice.htm
4. (PRACTICE) https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/law-sinescosines/e/law_of_cosines
Solving Triangles:
1.
(NOTES) http://www.mathsisfun.com/algebra/trig-solving-triangles.html
2.
(VIDEO-Right Triangles only) https://www.khanacademy.org/math/trigonometry/basictrigonometry/basic_trig_ratios/v/example--trig-to-solve-the-sides-and-angles-of-a-right-triangle
3. (PRACTICE PROBLEMS WITH SOLUTIONS WORKED OUT)
http://hotmath.com/help/gt/genericalg2/section_10_6.html
TRIGONOMETRY REVIEW (SOH CAH TOA)
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Find the measure of each unknown side.
Find the measure of each angle.
Law of Sines
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The LAW OF SINES is :
The Law of Sines is useful for finding:
TRY IT OUT ON THIS TRIANGLE:
USE THE LAW OF SINES TO SOLVE THE FOLLOWING TRIANGLES:
Law of Sines
Find each measurement indicated. Round your answers to the nearest tenth.
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Law of Sines
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Solve each triangle. Round your answers to the nearest tenth.
11) m∠A = 70°, c = 26, a = 25
15) m∠B = 117°, a = 16, b = 38
12) m∠B = 45°, a = 28, b = 27
16) m∠B = 84°, a = 18, b = 9
13) m∠C = 145°, b = 7, c = 33
17) m∠B = 105°, b = 23, a = 14
14) m∠B = 73°, a = 7, b = 5
18) m∠C = 13°, m∠A = 22°, c = 9
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APPLICATIONS OF LAW OF SINES
Boat Race: The course for a boat race starts at point A and
proceeds to point B, then to point C, and finally back to A.
Point C lies 8 kilometers directly south of point A.
Approximate the total distance of the race course.
Height of Clouds: To measure the height of clouds, a
spotlight is aimed vertically. Two observers at points A
and B, 364 feet apart and in line with the spotlight,
measure the two angles as shown in the diagram. How
far from the ground is the bottom of the cloud level?
OCEAN DEAPTHS: Two ships 1600 ft apart detect a submarine directly between them. The
angle of depression from the first ship to the submarine is 40°, and from the second ship is 28°.
How deep is the submarine?
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Law of COSines
The LAW OF COSINES is :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The Law of Cosines is useful for finding:

the third side of a triangle when you know …

the angles of a triangle when you know …
TRY IT OUT ON THIS TRIANGLE:
1.
Use the Law of Cosines to find the measure of….
Side c
2. Angle C
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Law of COSines
Find the measure indicated using the law of cosines.
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Law of COSines
Find the measure indicated using the law of cosines.
11.
12.
13.
14.
15.
16.
17.
18.
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APPLICATIONS OF LAW OF COSINES
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Review
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SOLVING TRIANGLES
To “SOLVE A TRIANGLE” means to….
In order to solve a triangle, you may need to use:
1.
3.
2.
EXAMPLES:
1.
A
4.
B
C
2.
3.
SOLVING TRIANGLES
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4.
5.
B
A
C
6.
7.
B
A
C
8.
SOLVING TRIANGLES
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9) In ∆ABC, a = 14 cm, b = 9 cm, c = 6 cm
10) In ∆XYZ, m∠X = 138°, y = 15 in, z = 25 in
11) In ∆QRP, q = 12 in, p = 28 in, r = 18 in
12) In ∆QRP, p = 28 km, q = 17 km, r = 15 km
13) In ∆DEF, e = 16 yd, d = 12 yd, f = 17 yd
14) In ∆RPQ, p = 18 mi, m∠R = 17°, q = 28 mi
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MIXED APPLICATIONS
1. Juan and Romella are standing at the seashore 10 miles apart. The coastline is a straight
line between them. Both can see the same ship in the water. The angle between the
coastline and the line between the ship and Juan is 35 degrees. The angle between the
coastline and the line between the ship and Romella is 45 degrees. How far is the ship from
Juan?
2. Jack is on one side of a 200-foot-wide canyon and Jill is on the other. Jack and Jill can both
see the trail guide at an angle of depression of 60 degrees. How far are they from the trail
guide?
3. Tom, Dick, and Harry are camping in their tents. If the distance between Tom and Dick is
153 feet, the distance between Tom and Harry is 201 feet, and the distance between Dick
and Harry is 175 feet, what is the angle between Dick, Harry, and Tom?
4. Three boats are at sea: Jenny one (J1), Jenny two (J2), and Jenny three (J3). The crew of J1
can see both J2 and J3. The angle between the line of sight to J2 and the line of sight to J3 is
45 degrees. If the distance between J1 and J2 is 2 miles and the distance between J1 and J3
is 4 miles, what is the distance between J2 and J3?
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MIXED APPLICATIONS
5. Airplane A is flying directly toward the airport which is 20 miles away. The pilot notices
airplane B 45 degrees to her right. Airplane B is also flying directly toward the airport. The
pilot of airplane B calculates that airplane A is 50 degrees to his left. Based on that
information, how far is airplane B from the airport?
6. A plane leaves JFK International Airport and travels due west at 570 mi/hr. Another plane
leaves 20 minutes later and travels 22º west of north at the rate of 585 mi/h. To the
nearest ten miles, how far apart are they 40 minutes after the second plane leaves.
7. Flights 104 and 217 are both approaching O’Hare International Airport from directions
directly opposite one another and at an altitude of 2.5 miles. The pilot on flight 104 reports
an angle of depression of 17º47’ to the tower, and the pilot on flight 217 reports an angle of
depression of 12º39’ to the tower. Calculate the distance between the planes.
8. Matt measures the angle of elevation of the peak of a mountain as 35º. Susie, who is 1200
feet closer on a straight level path, measures the angle of elevation as 42º. How high is the
mountain?
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MIXED APPLICATIONS
9. A triangular playground has sides of lengths 475 feet, 595 feet, and 401 feet. What are the
measures of the angles between the sides, to the nearest tenth of a degree?
10. A real estate agent has just take a trigonometry class at the local community college. She
is considering purchasing a piece of property and is waiting for the surveyor’s report before
closing the deal. If the surveyor submits a drawing as in the figure below, explain why the
agent will reject the sale.
400 ft
370 ft
71º
11. The surveyor admits to his mistake and revises his drawing as in the next figure. This
time the real estate agent refuses to complete the deal until additional information is
supplied. What additional information is the real estate agent looking for to complete her
knowledge about the parcel of land?
400 ft
370 ft
61º
12.
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GRAPHS OF TRIG FUNCTIONS
1. 𝑦 = sin 𝑥
x
y
0°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
360°
2. 𝑦 = cos 𝑥
x
y
0°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
360°
DOMAIN:
RANGE:
DOMAIN:
RANGE:
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3. 𝑦 = tan 𝑥
x
y
0°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
330°
360°
DOMAIN:
RANGE:
1. Compare the graphs of 𝑦 = sin 𝑥 and 𝑦 = cos 𝑥. How are the graphs similar?
How are the graphs different?
2. What is the value of the function 𝑦 = tan 𝑥 at x = 90° and x = 180° according
to your table? How is this shown on the graph?
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