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- WE WANT TO FIND THE TWO SMALLEST
POSITIVE RADIAN SOLUTIONS TO THE TRIG EQUATION
4 SINE 2 THETA = 3.
YOU MIGHT BE THINKING THAT WE HAVE TO PERFORM
A SUBSTITUTION FOR SINE 2 THETA, BUT WE'RE NOT GOING TO.
WE'RE GOING TO PERFORM A SUBSTITUTION JUST FOR 2 THETA.
WE'RE GOING TO LET BETA = 2 THETA,
WHICH WOULD GIVE US THE EQUATION 4 SINE BETA = 3.
SO WE'LL SOLVE THIS EQUATION FOR BETA AND THEN DETERMINE THETA.
NOTICE IF WE SOLVE THIS EQUATION FOR THETA WE WOULD HAVE
THETA = BETA DIVIDED BY 2.
SO WE'LL SOLVE THIS FOR BETA AND THEN DIVIDE BY 2
TO DETERMINE THETA.
SO TO SOLVE THIS EQUATION WE'LL FIRST DIVIDE BOTH SIDES BY 4.
SO WE HAVE SINE BETA = 3/4.
WELL, HAVING A SINE FUNCTION VALUE OF 3/4 IS NOT A VALUE
THAT WE'LL FIND ON THE UNIT CIRCLE
OR FROM REFERENCE TRIANGLES.
SO WE'LL HAVE TO USE THE CALCULATOR TO HELP
US SOLVE THIS EQUATION.
BUT BEFORE WE DO THAT, NOTICE HOW THE SINE FUNCTION
VALUE IS POSITIVE.
AND SINCE SINE THETA IS EQUAL TO Y DIVIDED BY R,
THIS IS TELLING US THAT THE Y COORDINATE
MUST BE POSITIVE IN ORDER FOR THE SINE FUNCTION VALUE
TO BE POSITIVE.
WHICH MEANS, THE ANGLES THAT SATISFY THIS EQUATION
MUST BE IN THE FIRST QUADRANT OR SECOND QUADRANT
WHERE Y IS POSITIVE.
NEXT, SINCE WE HAVE TO USE THE CALCULATOR TO HELP US FIND
ONE OF THE SOLUTIONS, IF SINE BETA = 3/4
THEN BETA IS GOING TO BE EQUAL TO ARC SINE
OR INVERSE SINE OF 3/4.
SO IF WE TYPE THIS INTO THE CALCULATOR
IT WILL GIVE US ONE SOLUTION FOR BETA.
SO LET'S GO AHEAD AND DO THAT.
BECAUSE WE WANT OUR ANSWER IN RADIANS, LET'S MAKE SURE
THAT WE ARE IN RADIAN MODE.
SO I'LL PRESS THE MODE KEY, NOTICE HOW RADIAN
IS ALREADY HIGHLIGHTED, SO WE ARE IN RADIAN MODE.
SO NOW WE'RE GOING TO PRESS SECOND SINE FOR INVERSE SINE,
AND THEN 3/4, CLOSED PARENTHESIS,
PRESS ENTER.
NOW, WE ARE ASKED TO ROUND OUR SOLUTION
TO THREE DECIMAL PLACES.
BUT THIS IS BETA NOT THETA, SO TO AVOID ANY ROUNDING ERRORS,
LET'S ROUND THIS TO FOUR DECIMAL PLACES.
SO ONE SOLUTION FOR BETA IS APPROXIMATELY 0.8481.
SO WE'LL CALL THIS BETA SUB 1.
LET'S GO AHEAD AND SKETCH THIS ANGLE
IN THE FIRST QUADRANT.
LET'S JUST SAY IT'S HERE.
WELL, SINCE THE SINE FUNCTION VALUE IS ALSO A POSITIVE
IN THE SECOND QUADRANT, IF WE SKETCH A REFERENCE ANGLE
IN THE SECOND QUADRANT OF 0.8481 RADIANS
IT'LL HAVE THE SAME SINE FUNCTION VALUE.
WHICH MEANS, THE ANGLE IN THE SECOND QUADRANT
WITH THE TERMINAL SIDE HERE, WITH THE REFERENCE ANGLE
OF 0.8481 RADIANS, WOULD ALSO BE A SOLUTION
TO OUR EQUATION.
SO HERE'S BETA SUB 1, AND THEN FOR BETA SUB 2,
ONE WAY TO FIND THIS ANGLE WOULD BE TO TAKE PI RADIANS
HALF A ROTATION, AND THEN SUBTRACT
THIS REFERENCE ANGLE TO DETERMINE THE MEASURE
OF THE SECOND SOLUTION.
SO LET'S GO AHEAD AND DO THAT.
WE WOULD HAVE PI RADIANS - 0.8481,
SO THE SECOND SOLUTION FOR BETA IN THE SECOND QUADRANT
IS APPROXIMATELY 2.2935 RADIANS.
BUT REMEMBER OUR GOAL IS TO FIND THETA NOT BETA.
AND SINCE THETA IS EQUAL TO BETA DIVIDED BY 2,
WE'LL NOW DIVIDE THESE BY 2 TO FIND OUR SOLUTIONS.
SO WE HAVE 0.8481 DIVIDED BY 2.
NOW WE WILL ROUND TO THREE DECIMAL PLACES,
SO THETA IS APPROXIMATELY 0.424.
AND NOW WE'LL FIND THE SECOND SOLUTION.
SO THETA IS APPROXIMATELY 1.147.
AND OF COURSE THESE ARE BOTH RADIANS.
THESE ARE THE TWO SMALLEST POSITIVE RADIAN SOLUTIONS
TO THE ORIGINAL TRIG EQUATION IN TERMS OF THETA.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.