- WELCOME TO ANOTHER VIDEO ON SOLVING TRIG EQUATIONS.
THE GOAL OF THIS VIDEO IS TO SOLVE TRIG EQUATIONS
USING A VARIETY OF STRATEGIES.
IF YOU LOOK AT THIS FIRST EQUATION
IT MAY BE DIFFICULT TO KNOW HOW TO APPROACH IT.
WE MAY TRY TO ADD THESE TWO FRACTIONS TOGETHER
BY OBTAINING A COMMON DENOMINATOR
BUT WHAT I THINK MIGHT BE HELPFUL
IS MULTIPLYING THIS SECOND FRACTION
BY THE CONJUGATE OF THE DENOMINATOR.
LET'S SEE WHAT THIS DOES.
SO THIS FIRST FRACTION WILL STAY THE SAME.
IN THE DENOMINATOR WE'RE GOING TO HAVE
ONE MINUS SINE SQUARED THETA
AND WE'LL LEAVE THE NUMERATOR IN FACTORED FORM.
NOW 1 - SINE SQUARED THETA IS = TO COSINE SQUARED THETA.
SO LET'S GO AHEAD AND DO THAT SUBSTITUTION NOW.
AND NOW THIS SECOND FRACTION SIMPLIFIES,
THERE'S A COMMON FACTOR OF COSINE THETA
SO THIS SIMPLIFIES OUT,
THIS WILL BE TO THE FIRST POWER.
AND NOW NOTICED THAT WE HAVE TWO FRACTIONS
WITH A COMMON DENOMINATOR
SO WE CAN GO AHEAD AND ADD THESE TWO FRACTIONS TOGETHER.
OUR DENOMINATOR WILL BE COSINE THETA
AND THE NUMERATOR WILL HAVE 1 + 1, THAT WOULD BE 2
AND NOTICE OUR SINE THETAS ALSO SIMPLIFY OUT.
WE HAVE SINE THETA PLUS A NEGATIVE SINE THETA
THAT WOULD BE ZERO.
SO WE HAVE 2/COSINE THETA = 4
AND THEN TO SOLVE THIS EQUATION FOR COSINE THETA
WE MAY WANT TO PERFORM CROSS PRODUCTS SO 4 COSINE THETA = 2
DIVIDING BY 4
ALL THIS COMES DOWN TO WHERE IS COSINE THETA EQUAL TO 1/2.
HAVING A COSINE FUNCTION VALUE OF 1/2
SHOULD REMIND YOU OF A 30-60-90 RIGHT TRIANGLE
WHERE THE COSINE OF A 60 DEGREE ANGLE IS EQUAL TO 1/2
THAT TELLS US OUR REFERENCE ANGLE WILL BE 60 DEGREES.
NEXT, COSINE THETA IS POSITIVE
IN BOTH THE FIRST AND THE FOURTH QUADRANT
SINCE COSINE THETA INVOLVES X
SO LET'S GO AHEAD AND SKETCH A 60 DEGREE REFERENCE ANGLE
IN THE FIRST AND THE FOURTH QUADRANTS.
THESE TWO ANGLES WILL HAVE A COSINE FUNCTION VALUE OF 1/2
THEREFORE THETA COULD BE 60 DEGREES OR PI/3 RADIANS,
OR THETA COULD ALSO BE 300 DEGREES OR 5 PI/3 RADIANS.
SO THIS EQUATION WAS FAIRLY INVOLVED
BUT REMEMBER ONE OF THOSE STRATEGIES
WHEN WE WERE USING TO VERIFY IDENTITIES
WAS TO MULTIPLY BY THE CONJUGATE.
LET'S TAKE A LOOK AT ANOTHER.
HERE WE HAVE SECANT SQUARED X -2 TAN X = 4.
WELL, WE CAN REPLACE SECANT SQUARED X WITH TAN SQUARED X +1,
LET'S GO AHEAD AND DO THAT.
WE HAVE AN EQUATION,
IT LOOKS LIKE IT'S GOING TO BE IN QUADRATIC FORM
SO LET'S GO AHEAD AND SET IT EQUAL TO ZERO
AND REARRANGE THE ORDER OF THESE TERMS.
IF WE SUBTRACT 4 ON BOTH SIDES WE'LL HAVE -3 ON THE LEFT.
AND LUCKILY, THIS IS FACTORABLE
TAN SCORED X IS A PERFECT SQUARE.
NOW WE CAN PLACE THE FACTORS OF -3 IN THE SECOND POSITIONS.
PUT A -3 HERE AND A +1 HERE.
OUR OUTER PRODUCT IS -3 TAN X, INNER PRODUCT IS +1 TAN X
WHICH DOES SUM TO THE MIDDLE TERM.
SO IF THIS PRODUCT IS = TO 0 THEN EITHER TAN X MUST = -1
OR TAN X MUST = 3.
HAVING A TANGENT FUNCTION VALUE OF -1
SHOULD REMIND US OF THE 45-45-90 RIGHT TRIANGLE
WHERE THE TANGENT OF 45 DEGREES IS = TO +1.
SO SINCE WE WANT A FUNCTION VALUE OF -1
WE HAVE TO DETERMINE WHICH QUADRANTS THE TANGENT FUNCTION
WOULD BE NEGATIVE.
IT WOULD BE NEGATIVE IN THE SECOND QUADRANT
SO WE'LL SKETCH A 45 DEGREE REFERENCE ANGLE HERE
AND IT WOULD ALSO BE NEGATIVE IN THE FOURTH QUADRANT.
REMEMBER, TANGENT IS Y/X.
SO HERE X COULD = 90 + 45 OR 135 DEGREES
OR 3 PI/4 RADIANS
OR 315 DEGREES WHICH WOULD BE 7 PI/4.
OKAY, FOR TAN X = 3 WE'RE GOING TO HAVE TO USE OUR CALCULATORS
BECAUSE THIS DOES NOT COME OUT TO A NICE REFERENCE ANGLE.
REMEMBER, IF WE SOLVE THIS EQUATION FOR X
BY TAKING THE INVERSE TANGENT OF BOTH SIDES
WE WOULD HAVE X = INVERSE TANGENT 3.
SO LET'S GO TO THE CALCULATOR NOW,
MAKE SURE WE'RE IN RADIAN MODE, WE ARE.
SO SECOND TANGENT BRINGS UP TO INVERSE,
IT LOOKS LIKE WE HAVE APPROXIMATELY 1.25 RADIANS,
GOING BACK TO OUR COORDINATE PLANE 1.25 RADIANS
WOULD BE IN THE FIRST QUADRANT,
WHILE TANGENT IS ALSO POSITIVE IN THE THIRD QUADRANT
SO IF WE SKETCH A REFERENCE ANGLE OF 1.25 RADIANS
WE CAN DETERMINE ANOTHER ANGLE IN THIS INTERVAL
THAT HAS A TANGENT FUNCTION VALUE OF 3.
SO THAT WOULD BE PI + 1.25 RADIANS
WHICH IS APPROXIMATELY 3.93.
SO IT LOOKS LIKE WE HAVE FOUR DIFFERENT SOLUTIONS
FOR THIS EQUATION.
I THINK WE HAVE TIME FOR ONE MORE.
OKAY, ON THIS LIST PROBLEM
WE HAVE TWO DIFFERENT TRIG FUNCTIONS
BOTH WITH DOUBLE ANGLES.
SO WHAT WE'LL TRY TO DO IS USE THESE DOUBLE ANGLE IDENTITIES
AND PERFORM SOME SUBSTITUTIONS TO MAKE THIS EASIER TO SOLVE.
LUCKILY, THERE'S ONLY ONE IDENTITY FOR SINE 2X
SO LET'S GO AHEAD AND REPLACE SINE 2X WITH 2 SIGN X COSINE X.
NOW FOR COSINE 2X
THERE'S THREE DIFFERENT IDENTITIES TO CHOOSE FROM.
THE ONE THAT STANDS OUT FOR ME IS THIS THIRD ONE
BECAUSE NOTICE WE HAVE COSINE 2X = 2,
COSINE SQUARED X -1,
NOTICE THERE'S A +1 IN THE ORIGINAL EQUATION
SO THOSE TWO WILL SIMPLIFY OUT IF I USE THIS IDENTITY.
SO LET'S GO AHEAD AND GIVE IT A TRY.
COSINE 2X IS EQUAL TO 2 COSINE SQUARED X - 1
AND THEN + 1 FROM THE ORIGINAL EQUATION.
SO THESE WILL GO OUT
AND NOW LET'S GO AHEAD AND TRY TO SET THIS EQUATION = TO 0.
NOW THIS IS FACTORABLE,
THERE'S A COMMON FACTOR OF 2 COSINE X,
LET'S TRY FACTORING THAT OUT.
THAT'S GOING TO LEAVE US WITH SINE X - COSINE X = 0.
NOW NORMALLY, THIS IS GOING TO BE A PROBLEM
HAVING TWO DIFFERENT TRIG FUNCTIONS
BUT LET'S GO AHEAD AND SEE WHAT WE CAN DO WITH IT.
SO WE'LL SET EACH OF THESE FACTORS EQUAL TO 0
DIVIDING BY 2 WE HAVE COSINE X = 0.
NOW, OVER HERE WHAT I'M GOING TO DO
IS MOVE THE COSINE X TO THE OTHER SIDE
AND NOW I'M GOING TO DIVIDE BOTH SIDES BY COSINE X.
REMEMBER, SINE X/COSINE X IS TAN X
AND OF COURSE, COSINE X DIVIDED BY COSINE X IS 1.
SO WE NEED TO FIND THE ANGLES ON THIS INTERVAL
WHERE COSINE X = 0 AND TAN X = 1.
REMEMBER ON THE UNIT CIRCLE COSINE THETA EQUALS X.
SO WE WANT THE POINTS ON THE UNIT CIRCLE
WHERE THE X-COORDINATE WOULD BE 0,
THOSE OCCUR ON THE Y-AXIS.
SO ON THIS INTERVAL WE WOULD HAVE PI/2 RADIANS
AND ALSO 3 PI/2.
NEXT, HAVING TAN X = TO 1 SHOULD REMIND US
OF A 45-45-90 RIGHT TRIANGLE
BECAUSE TANGENT 45 DEGREES IS = TO 1.
SO WHAT THAT TELLS US IS OUR REFERENCE ANGLE IS 45 DEGREES.
REMEMBER TANGENT IS POSITIVE IN THE FIRST QUADRANT
AND ALSO IN THE THIRD QUADRANT
SO WE'LL SKETCH A 45 DEGREE REFERENCE ANGLE HERE AS WELL.
SO THE FIRST ANGLE WOULD BE PI/4 RADIANS
AND THE SECOND ANGLE WOULD BE 225 DEGREES OR 5 PI/4 RADIANS.
SO WE HAD A COUPLE OF NON-ROUTINE EQUATIONS HERE,
HOPEFULLY THE EXPLANATIONS MADE SENSE
AND YOU FOUND THEM SOMEWHAT HELPFUL.
I HOPE YOU HAVE A GOOD DAY.