Sine & Cosine Law

Sine Law(Knowledge/Diagram)

100

According to sine law, what relationship is being described in a triangle? (20s)

Relationship between the sides and sine of their opposite angles

200

In the pictured triangle, ∠A is 98 degrees and ∠B is 12 degrees. If side a is 84 units long, approximately how long is side b?

18 units

300

Is it true that in the formula a/sinA = b/sinB = c/sinC (sine law), B is the angle opposite of side a?

No, B is the angle opposite of side b while A is the angle opposite of side a.

400

Triangle PQR has ∠P = 63.5° and ∠Q = 51.2° and r = 6.3 cm. What are the side lengths of p and q?

p = 6.21cm  

q = 5.40 cm

Sine Law(Word)

100

Diana and James are standing at the seashore 10 miles apart. The coastline is a straight line between them. Both can see the ship in the water. Than angle between the coastline and the line between the ship and Diana is 35 degrees. The angle between the coastline and the line between the ship and James is 45 degrees. How far is the ship from Juan?

 Find 3 angles with angle theorem: 35+ 45

                                                                                   180 - 80 = 100 degrees (third angle)

                            10/sin100 = x/sin45 

                            x=10sin45 divided by sin100 equals about 7.2 m

200

Fire towers A and B are located 10 miles apart. They use the direction of the other tower as 0°. Rangers at fire tower A spots a fire at 42°, and rangers at fire tower B spot the same fire at 64°. How far from tower A is the fire to the nearest tenth of a mile?

64 + 42 = 106

                              180-106 = 74 (third angle)

                              10/sin74 = x/sin64

                                          x = 10sin64 divided by sin74 equals about 9.3 or 9 miles.

300

Airplane A is flying directly toward an airport that is 20 miles away. The pilot notices airplane B 45 degrees to her right. Airplane B is also flying directly towards the airport. The pilot of airplane B calculates that airplane A is 50 degrees to his left. How far is airplane B from the airport?

20/sin50 = x/sin45

X = 20sin45 divided by sin50 equals about 18.5 miles.

400

A cottage under construction is to be 12.6 m wide. The two sides of the roof are to be supported by rafters that meet at an angle of 50 degrees. How long should the rafters be if they are the same length?

∠A + ∠C + 50 degrees = 180 degrees

                          2x + 50 degrees = 180 degrees

                               2x = 180 - 50 

                               2x = 130

                                 x = 65 degrees

                                a/sinA =b/sinB

                                a/sin65 = 12.6/sin50

                                A = 12.6sin65 divided by sin50 equals about 14.9 m

Cosine Law(Knowledge/Diagram)

100

What are the conditions we should have to solve a triangle with cosine law? (require 2 answers for points) (20s)

1. Two sides and a contained angle

2. All three sides

200

Solve x

19^2=8^2+14^2-2*8*14*cosX 

361= 64+196-224cosX

-101=-224cosX

cosX=101/224

<X=63.2

300

Ｗhat is the unknown angle shown in the diagram?

 2420^2=5050^2+6000^2-2*5050*6000*cos

        5856400=25502500+36000000-60600000*cos

        -55646100=-60600000*cos

        <=23.34

400

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 3½ feet above the ground. How far from vertical was the tree leaning?

 

 6^2=3.5^2+4^2-2*3.5*4*cos

    36=12.25+16-28cos

    7.75=-28cos

    <=106

    106-90=16

    A:16 ft

Cosine Law(Word)

100

Tom, Ricky, and Harry are camping in their tents. If the distance between Tom and Ricky is 153 ft, the distance between Tom and Harry is 201 ft, and the distance between Ricky and Harry is 175 ft, what is the angle between Ricky, Harry, and Tom?

153^2=201^2+175^2-2*201*175*cosH

    23409=40401+30625-70350*cosH

    -47617=-70350*cosH

    <H=47.4

200

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.?

180-38-47=95 

    a=11.5*2=23, b=13*2=26

    d^2=23^2+26^2-2*23*26*cos95

    d^2=529+676+104.238

    d=36.2

300

 To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake? (Hint: He walked the two sides of a triangle.)

 

    x^2=270^2+245^2-2*270*245*cos70

    x^2=72900+60025-45249.26

    x=296.1

400

Two ships leave the harbour at the same time. One ship travels on a bearing of S12°W at 14 miles per hour. The other ship travels on a bearing of N75°E at 10 miles per hour. How far apart will the ships be after three hours? Round to the nearest tenth of a mile.

x^2=30^2+42^2-2*30*42*cos117

 x=61.7