Applications of the Laws of Sines and Cosines Dr. Shildneck

HEADINGS Headings and Bearings are ways of measuring the direction someone or something is heading. Typical problems include planes and boats. Directions are typically given in one of two ways. (1) Headings (from North) When a direction is given as an angle between 0 o and 360 o, the direction is considered at the angle, clockwise from North. EXAMPLE (200 degrees) 1) Consider yourself (or the object) at the central location of a “compass.” 2) Begin by looking North. 3) Rotate clockwise, the number of degrees stated. 3) Go that direction. N

BEARINGS Bearings are given as a direction in reference to North or South. You are told to rotate some angle toward the east or west from that. Angles are between 0 o and 90 o. Notation for Bearings look like this. N(angle)E – Starting at north rotate toward East N(angle)W – Starting at north rotate toward West S(angle)E – Starting at north rotate toward East S(angle)W – Starting at north rotate toward West EXAMPLE (S20 o W degrees) 1) Consider yourself (or the object) at the central location of a “compass.” 2) Begin by looking North or South (indicated) 3) Rotate in the direction indicated the appropriate amount. 4) Go that direction. N

Example 1 A plane takes off from an airport heading N30 o E. After 100 miles it changes direction, now heading S70 o E. After another 100 miles, how far is the plane from where it took off?

Example 2 An 8-foot tall stop sign was hit by a car and is now leaning at an angle of 10 o from vertical. The angle of elevation of the sun is 40 o, how long is the sign’s shadow if the shadow is “inside” (on the same side of) the lean?

Example 3 A flag pole is on the side of a hill that makes a 10 o angle with the horizontal (level ground). The shadow of the flagpole is 20 feet long down the hill when the angle of elevation of the sun is 60 o. How tall is the flag pole?