- LET'S TAKE A LOOK AT AN APPLICATION PROBLEM THAT CAN BE SOLVED USING THE LAW OF SINES. IN THIS PROBLEM, WE HAVE A CELL TOWER THAT'S BEING INSTALLED ON A HILL. SO HERE'S THE CELL TOWER, AND THE HILL HAS AN ANGLE OF ELEVATION OF 42 DEGREES. AN ENGINEER HAS TO INSTALL A SUPPORT WIRE FOR THIS TOWER, AND THE QUESTION IS WHAT WOULD BE THE LENGTH OF THE SUPPORT WIRE IF IT'S GOING TO FORM A 15-DEGREE ANGLE WITH THE HILL. SO REMEMBER, TO APPLY THE LAW OF SINES, WE HAVE TO SET UP A PROPORTION WITH ONLY ONE UNKNOWN. SO WE HAVE TO HAVE AT LEAST A MEASURE OF ONE ANGLE AND THE LENGTH OF THE OPPOSITE SIDE TO MAKE THE LAW OF SINES WORK. AND RIGHT NOW, LOOKING AT THIS SMALL TRIANGLE ON THE HILL, WE DON'T HAVE ENOUGH INFORMATION. WE HAVE THE LENGTH OF THIS SIDE HERE, BUT NOT THE MEASURE OF THE OPPOSITE ANGLE. WE HAVE THE MEASURE OF THIS ANGLE HERE, BUT NOT THE LENGTH OF THE OPPOSITE SIDE. AND THEN WE DON'T KNOW THE MEASURE OF THIS ANGLE OR THE LENGTH OF THE OPPOSITE SIDE. SO LET'S GATHER ALL THE INFORMATION WE CAN ABOUT THIS TRIANGLE AND THEN SEE IF WE CAN SET UP THE LAW OF SINES TO SOLVE FOR X. THIS PROBLEM ACTUALLY RELIES ON A PROPERTY OF PARALLEL LINES FROM GEOMETRY. IF WE WERE TO SKETCH A SEGMENT HERE THAT WOULD BE PARALLEL TO THE GROUND, THEN WE CAN ASSUME THAT THIS ANGLE HERE WOULD BE 90 DEGREES. THEN SINCE THIS SEGMENT HERE AND THE GROUND HERE ARE PARALLEL, THE PARALLEL LINE IS CUT BY A TRANSVERSAL WHICH WOULD BE THIS SEGMENT HERE. THEN ALTERNATE INTO YOUR ANGLES ARE CONGRUENT WHICH MEANS THAT IF THIS ANGLE MEASURES 42 DEGREES, THEN THIS ANGLE HERE MUST ALSO BE 42 DEGREES WHICH TELLS US THAT THIS OBTUSE ANGLE AND THIS LITTLE TRIANGLE WOULD BE 90 DEGREES + 42 DEGREES OR 132 DEGREES. AND WE NEED THIS BECAUSE TO USE THE LAW OF SINES, WE'RE GOING TO HAVE THE SINE OF 132 DEGREES DIVIDED BY X AS ONE OF OUR RATIOS. AND NOW SINCE WE HAVE THE LENGTH OF THIS SIDE HERE AS 95 METERS, WE NEED TO DETERMINE THE MEASURE OF THIS ACUTE ANGLE HERE, AND WE CAN NOW BECAUSE WE KNOW THE SUM OF THE INTERIOR ANGLES OF ANY TRIANGLE IS 180 DEGREES. SO THE MEASURE OF THIS ANGLE HERE WOULD BE 180 DEGREES - 132 DEGREES - 15 DEGREES. SO THE MEASURE OF THIS ANGLE HERE MUST BE 33 DEGREES. AND NOW, WE CAN USE THE LAW OF SINES TO SOLVE FOR X. WE'RE GOING TO HAVE THE SINE OF 33 DEGREES DIVIDED BY THE LENGTH OF THE OPPOSITE SIDE WHICH IS 95 METERS MUST EQUAL THE SINE OF 132 DEGREES DIVIDED BY THE LENGTH OF THE OPPOSITE SIDE WHICH IS X. AND NOW, WE CAN CROSS MULTIPLY AND SOLVE FOR X. SO X x SINE 33 DEGREES MUST EQUAL 95 x SINE 132 DEGREES. NOW, WE'LL DIVIDE BOTH SIDES BY SINE 33 DEGREES. SO THE LEFT SIDE SIMPLIFIES TO X, AND X IS GOING TO BE EQUAL TO THIS QUOTIENT HERE. SO LET'S GO AHEAD AND EVALUATE THIS ON THE CALCULATOR. LET'S MAKE SURE WE'RE IN DEGREE MODE. NOTICE THAT WE ARE, SO OUR NUMERATOR IS 95 SINE 132 DEGREES DIVIDED BY SINE 33 DEGREES. SO X IS APPROXIMATELY 129.6 METERS WHICH SHOULD BE THE LENGTH OF THE SUPPORT WIRE IF IT'S GOING TO FORM A 15 DEGREE ANGLE WITH THE HILL. I HOPE THIS WAS HELPFUL.