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12.2 The Arc Functions

The trigonometric functions help us find the unknown length of the side of a right triangle if we know the  and the length of one side.  Sometimes weíll want to work backwards and find  if we know the lengths of the sides.  The arc functions allow us to do this.  With the arc functions , the relationships of  and the ratios are reversed.  In mathematical terms: 

                             (12- 12) 

The arc functions are often abbreviated with a -1 exponent.  For example:

Calculators usually determine the arc functions with an inverse key.  For example, rather than just hitting the "sin" key to calculate the sine of an angle, one would key in the ratio, hit the "INV" key and then the "sin" key to determine the angle.  Again, one must be conscious of whether the calculator is in a degrees or radians mode.

Example 12.7:  A quarterback throws an 18-yard pass to his wide receiver.  The receiver catches the ball 12 yards downfield from the quarterback.  At what angle with the yard markers was the pass thrown?

Solution:  The sine of the angle would be given by the equation:

           where  q = the angle between the pass and the yard markers 

Given the relationship of Equation 12-11:

By using a calculator:

The pass travels at a  angle with the yard markers.

The arctangent function serves a special purpose with vectors.  The arctangent function allows us to determine the angle of a vector directly from its i- and j-components.  Similar to the arcsin function, the arctangent is abbreviated as "arctan."  In mathematical terms: 

                               (12-13) 

In one of the example problems in the Vectors chapter, we studied a person rowing perpendicular to the flow of a river.  Since we know the velocity components of the rowing and the water current, we can use the arctangent function to determine the angle at which the boat will travel.

Example 12.8:  With Al rowing at 12 ft./sec and the river flowing at 5 ft./sec., at what angle downstream will he be traveling as he crosses the river?

Solution:  The resultant velocity vector is -5i + 12j.  The angle is determined by taking the arctan of .

By using a calculator, we see that  equals .  This means that the boat will veer  downstream