- WELCOME TO A SERIES OF VIDEOS
ON SOLVING TRIG EQUATIONS.
THE GOAL OF THIS VIDEO
IS TO SOLVE THE MOST BASIC TYPE OF TRIG EQUATIONS.
JUST LIKE SOLVING ALGEBRAIC EQUATIONS,
THERE ARE SEVERAL METHODS USED TO SOLVE TRIG EQUATIONS.
IT TAKES PRACTICE IN RECOGNIZING
WHICH TECHNIQUE TO USE WHEN SOLVING TRIG EQUATIONS.
THIS VIDEO WILL EXPLAIN
HOW TO SOLVE TRIG EQUATIONS IN LINEAR FORM
WITH ONE TRIG FUNCTION.
AND THERE'LL BE SEVERAL OTHER VIDEOS
THAT ADDRESS DIFFERENT TYPES OF EQUATIONS
WITH DIFFERENT SOLVING TECHNIQUES.
WE WANT TO SOLVE EACH EQUATION FIRST ON THE INTERVAL
FROM ZERO TO 2PI
AND THEN OVER ALL RADIAN MEASURE.
SO WHAT WE'RE GOING TO DO FIRST
IS SOLVE THIS EQUATION FOR SINE THETA.
SO WE'LL ADD TO BOTH SIDES AND THEN DIVIDE BY 2.
SO SINE THETA MUST EQUAL 1/2.
FIRST WE WANT TO FIND ALL THE ANGLES
THAT HAVE A SINE FUNCTION VALUE OF 1/2
ON THE INTERVAL FROM ZERO TO 2PI.
NOW, WE COULD USE REFERENCE TRIANGLES,
BUT IN THIS CASE I'M GOING TO GO AHEAD
AND USE THE UNIT CIRCLE.
REMEMBER, THE Y COORDINATE ON THE UNIT CIRCLE
IS EQUAL TO SINE THETA.
SO FIRST, IF WE KNOW THAT THE Y COORDINATE WILL BE +1/2
OUR ANGLES MUST BE IN EITHER THE FIRST OR SECOND QUADRANT.
SO WE'RE LOOKING FOR A Y COORDINATE OF 1/2,
AND RIGHT AWAY WE SEE ONE HERE,
30 DEGREES OR PI/6 RADIANS.
NOTICE WITH A 30 DEGREE REFERENCE ANGLE
IN THE SECOND QUADRANT
THAT PRODUCES AN ANGLE OF 150 DEGREES OR 5PI/6 RADIANS.
AND, AGAIN, THE Y COORDINATE IS EQUAL TO 1/2,
THEREFORE, SINE THETA IS EQUAL TO 1/2.
SO WE HAVE TWO ANGLES ON THE INTERVAL FROM ZERO TO 2PI
THAT PRODUCE A SINE FUNCTION VALUE OF 1/2.
SO THAT TAKES CARE OF THE INTERVAL FROM ZERO TO 2PI.
NOW, IF WE WANT TO FIND ALL RADIAN SOLUTIONS,
REMEMBER THAT THE PERIOD FOR THE SINE FUNCTION
IS 2PI RADIANS.
SO IF WE TAKE THIS ANGLE
AND ADD ANY MULTIPLE OF 2PI RADIANS
WE'LL HAVE ANOTHER COTERMINAL ANGLE WITH THIS ANGLE
AND, THEREFORE, IT'LL HAVE THE SAME SINE FUNCTION VALUE.
SO THETA COULD EQUAL PI/6 RADIANS PLUS--
WE'LL SAY 2PI K WHERE K IS SUM INTEGER.
SO THIS WILL PRODUCE SUM MULTIPLE OF 2PI RADIANS.
OR THETA COULD ALSO BE 5PI/6 RADIANS + 2PI K.
AND THAT WOULD PRODUCE ANY COTERMINAL ANGLE
WITH THIS TERMINAL SIDE
PRODUCING A SINE FUNCTION VALUE OF 1/2 AS WELL.
LET'S LOOK AT IT GRAPHICALLY NOW.
HERE WE HAVE THE GRAPH OF Y = SINE THETA AND THEN Y = 1/2.
THESE INTERSECTION POINTS REPRESENT SOLUTIONS.
SO THE FIRST TWO SOLUTIONS WE FOUND FROM ZERO TO 2PI
WERE HERE AND HERE.
THESE OTHER POINTS OF INTERSECTIONS
WOULD BE OBTAINED BY TAKING ONE OF THESE ANGLES
AND ADDING A MULTIPLE OF 2PI RADIANS.
THE DISTANCE FROM THIS POINT TO THIS POINT IS 2PI RADIANS
AND THE DISTANCE FROM THIS POINT TO THIS POINT
IS ANOTHER 2PI RADIANS.
SO YOU KIND OF SEE
WHAT'S HAPPENING HERE GRAPHICALLY AS WELL.
HERE WE HAVE ANOTHER EQUATION,
LET'S GO AHEAD AND SOLVE THIS FOR COSINE THETA.
SO WE'LL SUBTRACT THE SQUARE ROOT OF 2 ON BOTH SIDES,
THEN DIVIDE BY 2.
COSINE THETA MUST EQUAL -SQUARE ROOT 2/2.
WELL, AGAIN, USING A UNIT CIRCLE,
COSINE THETA IS EQUAL TO X.
SO NOW WE'LL LOOK FOR AN X COORDINATE
OF -SQUARE ROOT 2/2.
AND THAT'S GOING TO LIMIT US
TO THE SECOND AND THIRD QUADRANT
SINCE THE X COORDINATE IS GOING TO BE NEGATIVE.
SO LOOKING IN THE SECOND QUADRANT,
HERE'S AN X COORDINATE OF -SQUARE ROOT 2/2.
THE ANGLE IS 135 DEGREES OR 3PI/4 RADIANS.
AND IN THE THIRD QUADRANT,
HERE'S THE COSINE VALUE OF - SQUARE ROOT 2/2,
WHICH IS 5PI/4 RADIANS.
AND, AGAIN, TO FIND ALL RADIAN SOLUTIONS
WE'LL JUST HAVE TO ADD MULTIPLES OF 2PI RADIANS
TO BOTH OF THESE.
SO TO SOLVE THESE TYPES OF EQUATIONS
WITH SINE AND COSINE THE UNIT CIRCLE COMES IN VERY HANDY.
LET'S TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
HERE'S THE GRAPH OF Y = COSINE THETA
AND WE HAVE THE GRAPH OF Y = - SQUARE ROOT 2/2.
THE SOLUTIONS THAT WE FOUND ON THE INTERVAL FROM ZERO TO 2PI
WERE HERE AND HERE.
AND THESE OTHER SOLUTIONS WE SEE
WOULD BE THESE VALUES PLUS OR MINUS MULTIPLES OF 2PI.
OKAY, LET'S TAKE A LOOK AT AN EQUATION NOW
THAT INVOLVES TANGENT THETA.
AGAIN, WE'LL FIRST SOLVE THIS FOR TAN THETA
BY ADDING 1 TO BOTH SIDES, DIVIDING BY SQUARE ROOT 3.
SO WE HAVE TAN THETA = 1/SQUARE ROOT 3.
NOW, THIS RATIO SHOULD REMIND YOU
OF 30, 60, 90 RIGHT TRIANGLE.
IF YOU TAKE A LOOK AT A 30 DEGREE ANGLE,
THE TANGENT RATIO
OR THE OPPOSITE SIDE OVER THE ADJACENT SIDE
IS 1/SQUARE ROOT 3.
SO THAT GIVES US OUR FIRST MEASURE FOR ANGLE THETA
AS 30 DEGREES OR PI/6 RADIANS.
LET'S ALSO SKETCH THIS ON THE COORDINATE PLANE.
NOW, REMEMBER THAT TANGENT THETA IS ALSO POSITIVE
IN A THIRD QUADRANT
WHERE BOTH THE X AND THE Y COORDINATES ARE NEGATIVE.
SO IF WE SKETCH A 30 DEGREE REFERENCE ANGLE
IN THE THIRD QUADRANT
THIS ANGLE WOULD ALSO HAVE A TANGENT FUNCTION VALUE
OF 1/SQUARE ROOT 3.
AND THIS ANGLE WOULD BE 180 + 30 OR 210 DEGREES,
WHICH IS EQUAL TO 7PI/6 RADIANS.
NOW, FOR ALL RADIAN SOLUTIONS,
REMEMBER THE PERIOD FOR TANGENT THETA
IS PI RADIANS OR 180 DEGREES.
SO FOR THIS PROBLEM WE'RE ONLY GOING TO HAVE
ONE EQUATION FOR THETA AND IT WILL EQUAL--
WE CAN USE EITHER OF THESE ANGLES,
BUT LET'S USE PI/6 PLUS ANY MULTIPLE OF PI RADIANS.
SO WE'LL GET THIS SECOND ANGLE AND ANY OTHER COTERMINAL ANGLE
BY ADDING MULTIPLES OF PI TO PI/6 RADIANS.
SO WE DON'T NEED A SECOND EQUATION HERE
FOR ALREADY IN MEASURE DUE TO THE PERIOD BEING PI
INSTEAD OF 2PI.
LET'S LOOK AT IT GRAPHICALLY.
HERE ARE THE TWO SOLUTIONS WE LISTED FROM ZERO TO 2PI,
BUT YOU CAN SEE ALL OTHER SOLUTIONS
WILL BE MULTIPLES OF PI FROM EITHER OF THESE.
WE CHOSE PI/6 FOR OUR EQUATION.
LET'S TAKE A LOOK AT ANOTHER EQUATION.
4 COSINE THETA - 6 = COSINE THETA.
WELL, IN ORDER TO SOLVE THIS FOR COSINE THETA
WE MUST GET COSINE THETA ON THE SAME SIDE.
SO WE'LL SUBTRACT COSINE THETA, ADD 6 TO BOTH SIDES,
THEN DIVIDE BY 3, COSINE THETA = 2.
REMEMBER THE RANGE FOR COSINE THETA
IS ACTUALLY THE CLOSED INTERVAL FROM -1 TO +1.
SO COSINE THETA WILL NEVER EQUAL 2,
THEREFORE, FOR THIS EQUATION WE HAVE NO SOLUTION.
LET'S GO AND TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
HERE WE HAVE THE GRAPH OF Y = 2
AND THERE'S Y = COSINE THETA.
SINCE THEY DON'T INTERSECT
THIS VERIFIES OUR ANSWER OF NO SOLUTION.
OKAY. I HOPE YOU HAVE FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING
AND REMEMBER THIS IS JUST THE FIRST OF SEVERAL VIDEOS
ON SOLVING TRIG EQUATIONS.