- WELCOME TO THE THIRD 
OF A SERIES OF VIDEOS
ON HOW TO SOLVE 
TRIG EQUATIONS.
IN THIS VIDEO WE'LL TALK ABOUT 
HOW TO SOLVE TRIG EQUATIONS
THAT REQUIRE THE USE 
OF IDENTITIES,
HOW TO SOLVE TRIG EQUATIONS 
THAT ARE NOT FACTORABLE,
AND ALSO HOW TO SOLVE 
TRIG EQUATIONS
BY SQUARING BOTH SIDES.
SO THE FIRST TWO VIDEOS 
ARE PRETTY STRAIGHTFORWARD.
NOW WE'RE GOING TO SEE SOME 
MORE CHALLENGING PROBLEMS.
ON THIS FIRST PROBLEM 
WE WANT TO SOLVE THE EQUATION
ON THE INTERVAL 
FROM ZERO TO 360 DEGREES
AND WE HAVE COSINE SQUARED X 
- SINE SQUARED X = 1/2.
NORMALLY WHEN WE CAN WRITE 
AN EQUATION
IN TERMS OF ONE TRIG FUNCTION 
THAT'S VERY HELPFUL.
SO REMEMBER THE IDENTITY SINE 
SQUARED X + COSINE SQUARED X
= 1.
WE HAVE TO MAKE A CHOICE
WHETHER WE WANT TO SUBSTITUTE 
AN EXPRESSION IN
FOR COSINE SQUARED X 
OR SINE SQUARED X.
AND IN MANY CASES THERE'S 
MORE THAN ONE WAY TO DO THIS,
BUT I'M GOING TO REPLACE 
SINE SQUARED X
WITH 1 - COSINE SQUARED X.
AND SINCE WE'RE SUBTRACTING 
SINE SQUARED X
WE MUST SUBTRACT THE QUANTITY 
1 - COSINE SQUARED X.
NOW LET'S GO AHEAD 
AND CLEAR THESE PARENTHESIS.
NOTICE WE'LL HAVE - 1 AND THEN 
ALSO + COSINE SQUARED X.
SO WE'LL END UP HAVING 2 
COSINE SQUARED X - 1 = 1/2.
NEXT, WE'LL ISOLATE 
THE COSINE SQUARED X,
SO WE'LL ADD 1 TO BOTH SIDES.
1/2 + 1 WOULD BE 3/2.
AND NOW WE'LL DIVIDE BY 2 
OR MULTIPLY BY 1/2.
SO WE HAVE COSINE SQUARED X 
= 3/4.
NOW WE'LL SQUARE ROOT 
BOTH SIDES OF THE EQUATION.
SO WE HAVE COSINE X = + OR - 
THE SQUARE ROOT OF 3/2.
NOW, THIS RATIO SHOULD 
REMIND US
OF A 30, 60, 90 RIGHT TRIANGLE
WHERE THE COSINE 
OF A 30 DEGREE ANGLE
IS EQUAL TO SQUARE ROOT 3/2,
WHICH TELLS US OUR REFERENCE 
ANGLE WILL BE 30 DEGREES.
AND BECAUSE THE COSINE 
FUNCTION VALUE
IS + OR - SQUARE ROOT 3/2,
WE ARE GOING TO HAVE TO SKETCH 
A 30 DEGREE REFERENCE ANGLE
IN EACH QUADRANT 
TO DETERMINE ALL OF THE ANGLES
THAT WILL PRODUCE 
THIS COSINE FUNCTION VALUE.
SO LET'S GO AHEAD AND DO THAT.
LET'S GO AHEAD AND LABEL ALL 
OF THE SIDES OF EACH TRIANGLE.
AND NOTICE THAT EACH 
OF THESE ANGLES
WOULD HAVE A COSINE 
FUNCTION VALUE
OF + OR - SQUARE ROOT 3.
SO LET'S GO AHEAD 
AND FIGURE OUT
WHAT ALL THESE ANGLES 
WOULD BE.
THERE'S 30 DEGREES.
THE NEXT ANGLE WOULD BE 
180 - 30 OR 150 DEGREES.
THE THIRD ANGLE WOULD BE 
180 + 30 THAT'S 210.
AND LASTLY, THIS ANGLE HERE 
WOULD BE 360 - 30
OR 330 DEGREES.
LET'S GO AHEAD AND CHECK THIS 
ON THE GRAPHING CALCULATOR.
WHAT WE CAN DO IS TYPE THE 
LEFT SIDE OF THE EQUATION
INTO Y1,
AND THE RIGHT SIDE INTO Y2,
AND THEN USE THE TABLE FEATURE 
TO CHECK.
SO WHAT WE'LL DO NOW 
IS MAKE SURE
THAT Y1 AND Y2 ARE EQUAL 
AT THESE VALUES OF X.
LET'S MAKE SURE WE'RE 
IN DEGREE MODE, WHICH WE ARE.
SO NOW WE'LL GO 
TO OUR TABLE SECOND GRAPH.
LET'S ALSO MAKE SURE 
THAT WE HAVE OUR TABLE
ON THE ASK OPTION.
SO WE'LL GO DOWN 
TO THE INDEPENDENT VARIABLE,
MAKE SURE IT'S ON ASK, 
WHICH IT IS.
PRESS 2nd, GRAPH,
AND NOW WE'LL TYPE IN WHAT 
WE THINK THE SOLUTIONS ARE.
AND IT ALL LOOKS GOOD
BECAUSE Y1 IS EQUAL TO Y2 
FOR THESE X VALUES.
LET'S GO AHEAD 
AND TRY ANOTHER PROBLEM.
NOW, FOR THIS PROBLEM THERE'S 
NO OBVIOUS SUBSTITUTION,
HOWEVER, THIS SHOULD REMIND US 
OF THE IDENTITY
TAN SQUARE THETA + 1 
= SECANT SQUARED THETA.
SO WHAT WE CAN DO HERE,
IS ACTUALLY SQUARE 
BOTH SIDES OF THE EQUATION,
AND WHEN WE DO THAT WE WILL 
OBTAIN A TAN SQUARED THETA
AND ALSO A SECANT SQUARED 
THETA.
AND THEN WE CAN USE 
THIS IDENTITY
TO PERFORM A SUBSTITUTION.
SO LET'S GO AHEAD 
AND TRY THAT.
WE'LL SQUARE BOTH SIDES.
NOW, WE'D ACTUALLY HAVE 
TO FOIL THIS OUT,
THE RESULT WOULD BE 
TAN SQUARED THETA
+ 2 SQUARE ROOT 3 TAN THETA 
+ 3.
NOW, REMEMBER THIS MIDDLE TERM 
CAME FROM FOILING THIS OUT,
WHICH WE'D WRITE THIS OUT 
TWICE AND THEN FOIL IT.
NOW, TO GET THIS EQUATION 
IN TERMS OF JUST TANGENT,
WE CAN NOW REPLACE 
SECANT SQUARED THETA
WITH TAN SQUARED THETA + 1.
SO WE'LL DO A SUBSTITUTION 
HERE.
AND NOW THAT WE HAVE 
EVERYTHING
IN TERMS OF TANGENT,
WE CAN SEE THAT IF WE SUBTRACT 
TAN SQUARED THETA
FROM BOTH SIDES THESE 
TWO TERMS WILL BE ELIMINATED.
SO NOW WE CAN SOLVE 
THIS EQUATION FOR TAN THETA.
SO THE FIRST STEP WOULD BE 
TO SUBTRACT 3 ON BOTH SIDES.
SO WE'D HAVE 2 SQUARE ROOT 3 
TANGENT THETA = -2,
DIVIDE BY 2 SQUARE ROOT 3,
SO WE'RE LEFT 
WITH TANGENT THETA
= THIS SIMPLIFIES 
TO - 1/SQUARE ROOT 3.
SO WE'RE LOOKING 
FOR THE ANGLES
THAT HAVE A TANGENT FUNCTION 
VALUE OF -1/SQUARE ROOT 3.
AND, AGAIN, THIS SHOULD 
REMIND US
OF A 30, 60, 90 
RIGHT TRIANGLE.
IF YOU TAKE A LOOK 
AT A 30 DEGREE ANGLE,
THE TANGENT OF THIS ANGLE 
WOULD BE 1/SQUARE ROOT 3.
SINCE WE HAVE 
A NEGATIVE FUNCTION VALUE,
WE KNOW THAT TANGENT 
IS NEGATIVE
IN THE SECOND AND FOURTH 
QUADRANT.
SO WHAT WE'LL DO,
IS WE'LL SKETCH A 30 DEGREE 
REFERENCE ANGLE
IN THE SECOND AND FOURTH 
QUADRANT
TO FIGURE OUT WHAT ANGLES.
SO THIS FIRST ANGLE LOOKS 
LIKE IT WOULD BE 150 DEGREES
AND THE SECOND SOLUTION LOOKS 
LIKE IT WOULD BE 360 - 30
OR 330 DEGREES.
AND THEY DO ASK FOR THE ANGLES 
TO BE IN RADIANS,
SO 150 DEGREES 
WOULD BE 5PI/6 RADIANS
AND 330 DEGREES 
WOULD BE 11PI/6 RADIANS.
NOW, WE HAVE TO BE CAREFUL 
HERE.
WHEN YOU SQUARE BOTH SIDES 
OF AN EQUATION
SOMETIMES YOU GET EXTRANEOUS 
SOLUTIONS,
WHICH ARE SOLUTIONS 
THAT MAY APPEAR AS SOLUTIONS,
BUT THEY DON'T ACTUALLY WORK 
IN THE ORIGINAL EQUATION.
SO LET'S GO AHEAD 
AND CHECK THESE
LIKE WE DID ON NUMBER ONE.
SO WE'LL TYPE THE LEFT SIDE 
IN Y1,
AND THEN IN Y2 
WE'LL TYPE SECANT X.
THERE'S NO SECANT KEY 
SO WE'LL TYPE IN 1/COSINE X.
GO TO OUR TABLE 
AND WE'LL TYPE IN
WHAT WE THINK 
OUR SOLUTIONS ARE.
I'M ALREADY IN DEGREE MODE
SO I'M GOING TO TYPE IN 
150 AND 330.
WHAT WE'LL NOTICE HERE 
IS THAT 150 DEGREES,
Y1 AND Y2 ARE OPPOSITE, 
SO THEY'RE NOT EQUAL.
SO 150 DEGREES 
IS NOT A SOLUTION.
AND NOTICE THAT 330 DEGREES 
DOES CHECK.
SO IT IS IMPORTANT 
THAT WE CHECK OUR SOLUTIONS
WHEN WE SQUARE BOTH SIDES 
OF AN EQUATION.
THIS IS NOT A SOLUTION.
WE HAVE ONE SOLUTION, WHICH 
IS 11PI/6 OR 330 DEGREES.
OKAY. I THINK WE HAVE TIME 
FOR ONE MORE.
SO THERE'S TWO POSSIBLE 
SUBSTITUTIONS
WE COULD DO HERE.
WE COULD TRY TO REPLACE
SINE SQUARED X WITH 1 
- COSINE SQUARED X,
BUT THEN WE STILL HAVE 
THIS DOUBLE ANGLE.
SO WE'RE GOING TO GO AHEAD 
AND REPLACE COSINE 2X
WITH COSINE SQUARED X 
- SINE SQUARED X
AND SEE WHERE THAT LEADS US.
AND THIS DOES HELP OUT
BECAUSE NOTICE WE HAVE - SINE 
SQUARED X + SINE SQUARED X,
SO THOSE ADD TO ZERO.
AND WE'RE LEFT WITH COSINE 
SQUARED X - 3 COSINE X = 1.
LOOKS LIKE IT'S 
IN QUADRATIC FORM,
SO LET'S GO AHEAD 
AND SET IT EQUAL TO ZERO.
NOW, THE OTHER PROBLEMS 
THAT WE'VE SEEN--
THIS WAS ALWAYS FACTORABLE,
BUT NOW WE HAVE COME ACROSS 
ONE THAT'S IN QUADRATIC FORM
BUT NOT FACTORABLE.
SO WE'RE GOING TO HAVE TO USE 
THE QUADRATIC FORMULA
TO FIND THE POSSIBLE SOLUTIONS 
FOR THIS
WHERE OUR "A" VALUE WILL 
EQUAL 1, B = -3, AND C = -1.
THE DIFFERENCE HERE 
IS INSTEAD OF SAYING X EQUALS
WE'RE GOING TO HAVE COSINE X 
= -B OR -3 + OR -
THE SQUARE ROOT OF B SQUARED 
- 4AC/2 x "A."
AND THIS WILL SIMPLIFY TO 3 
+ OR - THE SQUARE ROOT OF 9
+ 4 SQUARE ROOT OF 13/2.
LET'S GO AHEAD AND FIND 
THE DECIMAL APPROXIMATIONS
OF THESE TWO VALUES.
3 + THE SQUARE ROOT OF 13 
DIVIDED BY 2
IS APPROXIMATELY 3.3028.
AND 3 - SQUARE ROOT 13 DIVIDED 
BY 2 IS APPROXIMATELY -0.3028.
THE FIRST THING 
WE SHOULD RECOGNIZE
IS THAT THE RANGE FOR COSINE X
IS THE CLOSED INTERVAL 
FROM - 1 TO +1.
SO THIS WILL NOT GIVE US 
ANY POSSIBLE SOLUTIONS FOR X.
LET'S TAKE A CLOSER LOOK AT 
THIS ONE ON THE NEXT SCREEN.
WHAT WE CAN DO NOW IS TAKE THE 
INVERSE COSINE ON BOTH SIDES.
SO ON THE LEFT SIDE 
WE'LL HAVE X
AND ON THE RIGHT SIDE 
WE'LL GO TO OUR CALCULATOR,
AND THIS GIVES US 
APPROXIMATELY 107.6 DEGREES.
SKETCH THIS ANGLE 
ON THE COORDINATE PLANE.
SO WE'D HAVE 90 + ANOTHER 17.6 
DEGREES.
NOW, NOTICE THAT MEANS 
THAT OUR REFERENCE ANGLE HERE
WOULD BE 180 - 107.6 
OR 72.4 DEGREES.
THE REASON THAT'S IMPORTANT
IS BECAUSE REMEMBER 
THAT COSINE IS NEGATIVE
IN BOTH THE SECOND 
AND THE THIRD QUADRANT.
SO THERE'S ANOTHER ANGLE 
THAT HAS THE SAME
COSINE FUNCTION VALUE 
IN THE THIRD QUADRANT
WITH THE SAME REFERENCE ANGLE.
SO IF WE SKETCH 
ANOTHER REFERENCE ANGLE
OF 72.4 DEGREES 
IN THIS QUADRANT
THIS ANGLE HERE 
WILL BE ANOTHER SOLUTION
TO THIS EQUATION.
SO WE'D HAVE 180 + 72.4,
WHICH IS APPROXIMATELY 
252.4 DEGREES.
I HOPE YOU HAVE FOUND 
THIS VIDEO HELPFUL.
