First, draw the triangle by the following information: From Mount Crumpet you will go 200 miles at a bearing of N 20° E to Santa’s Sleigh Repair Shop, this distance is called a. From there you will go at a bearing of N 35° W for 40 miles to Santa’s Magical Workshop, this distance is b. From there you will return to Mount Crumpet, this distance is c. Opposite each side is an angle corresponding to each name only capitalized.

 Second, find angle C. Angle C is equal to the sum of the complement to the alternate interior angle of the 20° course bearing plus the complement of the 35° course bearing.

 90°-20°=70°           90°-35°=55°                 70°+55°=125°=Angle C

 Now, you can find the distance of side c by using the Law of Cosines. Remember, the Law of Cosines states:

c²=a²+b²-(2abCosC)

Use the given distances of a and b, 200 and 40, and the measurement you found for angle C, 125°, and plug them into the equation.

            c²=200²+40²-(2(200)(40)Cos125°)            

*Remember to take the square root when solving for c!

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c=225.34 miles

*We will round to the nearest hundredth throughout the problem

     Next, find the measure of angle B using the Law of Sines, which is:

            SinA/a=SinB/b=SinC/c

Plug the values you found for angle C and side c into the equation and the given value for side b.

 Sin125°/225.34=SinB/40            *Cross multiply and divide

 40Sin125°/225.34=SinB               *Remember to take the Sin inverse to find B!

       B=8.36°

            Now, you can find angle A by subtracting the measures of angle B and angle C from 180° (the three angles in a triangle add up to 180°).

            Angle A=180°-125°-8.36°=46.64°

     You also need to find the course bearing back from Santa’s Magical Workshop to Mount Crumpet.

            90°+55°+46.64°=191.64°

Now, subtract 180° from this measurement to find the degrees SW.

            191.64°-180°=11.64°        Write this as S 11.64° W

            Last, figure out how much fuel you will need for the hot air balloon flight.  First, figure out how long you will be in the air, based on the air speed and the distance. 

            *Use the formula d/r=t

            200 miles/80mph=2.5 hours

Now, multiply the number of hours by the amount of fuel burned per hour.

            2.5 hours x 15 gallons of propane/hour = 37.5 gallons of propane

This is how much fuel you will need for your flight. 

Congratulations!  You have solved for all necessary unknown data.  Now it is time to decide:

WILL YOU ACCEPT THIS MISSION?

OTHER TRIGONOMETRY LINKS

an introduction to trigonometry (very basic and factual)

trigonometry realms (information and neat animations)

trigonometry: university of guelph (information with diagrams)

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