Method 3 Law of Sines and Cosines Also

Steps § § Draw a rough sketch of the vectors The resultant is determined

These Laws Work for Any Triangle. A + B + C = 180° C

Example 2: using method 3 Stan is trying to rescue Kyle from drowning. Stan

Example (using same problem) Stan is trying to rescue Kyle from drowning. Stan gets

Calculating: Magnitude: c 2 = a 2 + b 2 – 2 abcos. C

Use the Law of: ¡ Sines when you know: l l 2 angles and

Advantages and Disadvantages of the Analytical Method ¡ ¡ ¡ Does not require drawing

This completes Method Three! Keep up the good work! This is our last time

Another Problem Paul is on a railroad flat car which is moving east at

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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 Any Triangle. A + B + C = 180° C Law of sines: a sin A b a = b = c sin B sin C Law of cosines: B A c c 2 = a 2 + b 2 – 2 abcos C

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

Calculating: Magnitude: c 2 = a 2 + b 2 – 2 abcos. C = (6 m/s)2 + (4 m/s)2 – 2(6 m/s)(4 m/s)cos 130° = 82. 85 c = 9. 10 m/s Direction: sin C = c sin 130° = 9. 10 sin B = 0. 337 R = 19. 67° + 20° R = 9. 1 m/s @ 39. 7° N of E sin B b sin B 4 B = 19. 67° = 39. 67°

Use the Law of: ¡ Sines when you know: l l 2 angles and an opposite side 2 sides and an opposite angle ¡ Cosines when you know: l 2 sides and the angle between them

Advantages and Disadvantages of 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 Problem Paul 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. 0 o 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. 7 o (or 7. 7 o N of E)