Method #3: Law of Sines and Cosines

Also referred to as the analytical method.

Steps

Draw a rough sketch of the vectors The resultant is determined using:

Algebra Trigonometry Geometry

These Laws Work for AnyAny Triangle.

a

c

b

C

B A

A + B + C = 180°

Law of sines:

a = b = c sin A sin B sin C

Law of cosines:

c2 = a2 + b2 –2abcos C

Example 2: using method 3Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.

Example (using same problem)Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.

Calculating:

Magnitude:c2 = a2 + b2 – 2abcosC

= (6m/s)2 + (4m/s)2 – 2(6m/s)(4m/s)cos130°= 82.85 c = 9.10 m/s

Direction:sin C = sin B c bsin 130° = sin B 9.10 4sin B = 0.337 B = 19.67°R = 19.67° + 20° = 39.67°

R = 9.1 m/s @ 39.7° N of E

Use the Law of:

Sines when you know: 2 angles and an

opposite side 2 sides and an

opposite angle

Cosines when you know: 2 sides and the

angle between them

Advantages and Disadvantagesof the Analytical Method

Does not require drawing to scale.

More precise answers are calculated.

Works for any type of triangle if appropriate laws are used.

Can only add 2 vectors at a time.

Must know many mathematical formulas.

Can be quite time consuming.

This completes Method Three!

Keep up the good work! This is our last time in class to learn these.

problems #5, 6 due tomorrow

Another ProblemPaul is on a railroad flat car which is moving east at 20.0 m/s (Vcg = velocity of the car relative to the ground) . Paul walks on the flat car at 5.0 m/s @ 40.0o N of E as shown (Vpc = velocity of Paul relative to the car) . What is Paul’s velocity relative to the ground (Vpg = velocity of Paul relative to the ground)?

Vpg = 24 m/s @ 7.7o (or 7.7o N of E)