ACC Precalculus Name ______________________________________

Law of Sines Date _____________________ Block __________

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General Triangle Trigonometry: In this unit we will round all answers to the nearest tenth. Use given or

exact values whenever possible. Any calculator may be used; calculator needs to be in degree mode.

We will learn to solve oblique (non-right) triangles using the Law of sines and Law of Cosines.

The Law of Sines can be used to solve AAS, ASA, and SSA (special case) triangles.

Solve the triangles using the given information.

Example 1: InEFG , g = 4.56, G = 43o, and E = 57 o. Solve the triangle.

_____ _____

_____ _____

_____ _____

= =

= =

= =

E e

F f

G g

Example 2: InABC , a = 16, B = 80o, and C = 34 o. Solve the triangle.

_____ _____

_____ _____

_____ _____

= =

= =

= =

A a

B b

C c

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Facts we need to remember:

1. In a triangle, the sum of the interior angles is 180o.

2. No triangle can have 2 obtuse angles.

3. The sine function has a range of −1,1 .

4. If 0 sin 1 , then can lie in the first quadrant (acute angle) or second quadrant (obtuse angle).

Determine if the given information supports 1 unique triangle, 2 triangles that are not congruent, or 0

triangles. If 1 or 2 triangles exist, solve the triangle(s).

Example 3: A = 65o, a = 18, and b= 22

Example 4: A = 58o, a = 25, and b= 22

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Example 5: A = 22o, a = 20, and b= 32

Summary

First, draw the triangle with given information and determine if you have AAS, ASA, or SSA.

Exactly one triangle is formed when given information for AAS or ASA triangles.

Refer to the given diagram. In the case of SSA, given angle A1, side a1, and side a2, set up your triangle

as below. Try to draw angle A1 like the given information (not to scale but representative).

1. Find 2A using law of sines. If 2sin 1A , then angle

2A does not exist and no triangle is formed. If you try

to find the arcsine, the calculator will give a domain error.

2. If 20 sin 1A , then angle 2A is acute; at least 1

triangle is formed. Find the measure of angle 2A and

then you can solve for angle 3A and side 3a .

3. To determine if 2 triangles are formed, find 2 'A

where 2 2' 180A A= − . If 1 2 ' 180AA + , then 2

triangles can be found. Find angle 3 'A and side 3 'a .

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Example 6: In RST , R = 105o, r = 12, and t= 10. Determine the number of triangles that can be formed and solve accordingly.

Related topic: Find the area of a triangle using sine.

The area of any ABC is one half the product of the lengths of two sides and the sine of the included angle. Let K be the area of a triangle. Then

1sin

2=K bc A

1sin

2=K ab C

1sin

2=K ac B

Choose the formula needed depending on the given information.

Example 7: A university landscaping architecture department is designing a garden for a triangular area in a dormitory complex. Two sides of the garden, formed by the sidewalks in front of buildings A and B, measure 172 ft and 186 ft, respectively, and together form a 53o angle. The third side of the garden, formed by the sidewalk along Crossroads Avenue, measures 160 ft. What is the area of the garden to the nearest square foot?

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Law of Cosines

The law of cosines is used to solve triangles given two sides and the included angle (SAS) or given three sides (SSS). When given 3 sides of a triangle (SSS), you must find the angle opposite the largest side first! In any triangle ABC,

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

= + −

= + −

= + −

a b c bc A

b a c ac B

c a b ab C

When the included angle is 90o, the law of cosines reduces to the Pythagorean theorem.

Example 1: In ABC , a = 32, c = 48, and B = 125.2o. Solve the triangle.

_____ _____

_____ _____

_____ _____

= =

= =

= =

A a

B b

C c

Example 2: Solve RST , given r = 3.5, s = 4.7, and t = 2.8.

_____ _____

_____ _____

_____ _____

= =

= =

= =

R r

S s

T t

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Use Heron’s Formula to find the area of a triangle when given 3 sides (SSS).

Note: s is the “semi-perimeter.” Example 3: Find the area of a triangle with side 5, 8, and 10. Example 4: A university landscaping architecture department is designing a garden for a triangular area in a dormitory complex. Two sides of the garden, formed by the sidewalks in front of buildings A and B, measure 172 ft and 186 ft, respectively, and together form a 53o angle. The third side of the garden, formed by the sidewalk along Crossroads Avenue, measures 160 ft. What is the area of the garden to the nearest square foot?

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Accelerated Precalculus Law of Sines and Cosines Applications 1

Set up each problem by drawing a diagram or picture, then write and solve an appropriate equation.

Answers: 1.) (a) SW station is closer, (b) 1.5 km 2.) 21.8 km 3.) 795.9 mi 4.) 3.3o 5.) 2.6 mi

6.) 48.3 ft 7.) 22,825.4 ft2 8.) 43.8 mi 9.) 32.9 ft 10.) 26.7 ft

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Accelerated Precalculus Law of Sines and Cosines Applications 2

1. After the hurricane, the small tree in my neighbor’s yard was leaning. To keep it from falling, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3 ½ feet above the ground. How far from vertical was the tree leaning?

2. Two ships leave port at 4 p.m. One is headed at a bearing of N 38 E and is traveling at 11.5 miles per

hour. The other is traveling 13 miles per hour at a bearing of S 47 E. How far apart are they when dinner is served at 6 p.m.?

3. Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call

that is received by each station. The call to Station Able indicates that the ship is located 55N E ; the call

to Station Baker indicates that the ship is located 60 .S E A. How far is each station from the ship? B. If a helicopter capable of flying 200 miles per hour is dispatched form the nearest station to the ship, how

long will it take to reach the ship?

4. The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle

of 10 . A. How long should a guy wire be it if is to connect to the tope of the tower and be secured at a point on the

sloped side 100 feet from the base of the tower? B. How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a

point 100 feet from the base on the flat side?

5. Each of two legs of a stepladder is 12 feet long. If the angle formed by the legs measures 13 , how far apart are the feet of the stepladder?

6. Two planes, one flying at 300 miles per hour and the other at 450 miles per hour, left the same airport at noon. At 3pm they were 1200 miles apart. What was the measure of the angle between their flight paths?

7. A straight road makes an angle of 22 with the horizontal. From a certain point P on the road, the angle of

elevation of an airplane is 57 . At the same instant, from another point 100 meters father up the road,

the angle of elevation is 63 . How far is the airplane from point P?

8. The sides of a parallelogram have lengths 60 feet and 80 feet and the smallest angle has measure 56 . Find the area of the parallelogram and the length of each of the diagonals.

9. A string is tied in a loop and then pulled taut from three points in the string to make a triangle with sides 60 inches, 75 inches, and 100 inches. What are the measures of the angles of the triangle?

10. An airplane is sighted simultaneously from two towns that are 3 miles apart. The angles of elevation from

town A is 40.8 and the angle of elevation from town B is 75 . If the airplane is directly above a straight line between the two towns, how far is the airplane from each town?

Answers: 1. 16.07 2. 36.18 miles 3a. Able=143.33mi, Baker=135.58mi 3b. 41 min 4a. 492.6ft 4b. 269.3ft; 5. 2.72 feet

6. 60.61 7. 627.64m 8. diagonals = 68.05ft and 123.97ft, Area=3,979.38ft 2 9. 94.94 , 48.35 , 36.71 respectively

10. A=3.22 miles B=2.18miles

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ACC Pre-Calculus

Unit 5 Review Law of Sines and Cosines

Name: __________________________

Date: _______________ Block: _______

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11) 79.7 km 12) 2028.1 or 2028 miles 13) distance from point A = 1.6 mi, height = 1.0 mi

14) Possible distances are 35.2” and 63.7”