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00:05

- WELCOME TO A SERIES OF VIDEOS
- WELCOME TO A SERIES OF VIDEOS
00:06

ON SOLVING TRIG EQUATIONS.
ON SOLVING TRIG EQUATIONS.
00:09

THE GOAL OF THIS VIDEO
THE GOAL OF THIS VIDEO
00:09

IS TO SOLVE THE MOST BASIC TYPE OF TRIG EQUATIONS.
IS TO SOLVE THE MOST BASIC TYPE OF TRIG EQUATIONS.
00:14

JUST LIKE SOLVING ALGEBRAIC EQUATIONS,
JUST LIKE SOLVING ALGEBRAIC EQUATIONS,
00:16

THERE ARE SEVERAL METHODS USED TO SOLVE TRIG EQUATIONS.
THERE ARE SEVERAL METHODS USED TO SOLVE TRIG EQUATIONS.
00:19

IT TAKES PRACTICE IN RECOGNIZING
IT TAKES PRACTICE IN RECOGNIZING
00:20

WHICH TECHNIQUE TO USE WHEN SOLVING TRIG EQUATIONS.
WHICH TECHNIQUE TO USE WHEN SOLVING TRIG EQUATIONS.
00:24

THIS VIDEO WILL EXPLAIN
THIS VIDEO WILL EXPLAIN
00:25

HOW TO SOLVE TRIG EQUATIONS IN LINEAR FORM
HOW TO SOLVE TRIG EQUATIONS IN LINEAR FORM
00:27

WITH ONE TRIG FUNCTION.
WITH ONE TRIG FUNCTION.
00:29

AND THERE'LL BE SEVERAL OTHER VIDEOS
AND THERE'LL BE SEVERAL OTHER VIDEOS
00:30

THAT ADDRESS DIFFERENT TYPES OF EQUATIONS
THAT ADDRESS DIFFERENT TYPES OF EQUATIONS
00:32

WITH DIFFERENT SOLVING TECHNIQUES.
WITH DIFFERENT SOLVING TECHNIQUES.
00:35

WE WANT TO SOLVE EACH EQUATION FIRST ON THE INTERVAL
WE WANT TO SOLVE EACH EQUATION FIRST ON THE INTERVAL
00:37

FROM ZERO TO 2PI
FROM ZERO TO 2PI
00:40

AND THEN OVER ALL RADIAN MEASURE.
AND THEN OVER ALL RADIAN MEASURE.
00:43

SO WHAT WE'RE GOING TO DO FIRST
SO WHAT WE'RE GOING TO DO FIRST
00:44

IS SOLVE THIS EQUATION FOR SINE THETA.
IS SOLVE THIS EQUATION FOR SINE THETA.
00:46

SO WE'LL ADD TO BOTH SIDES AND THEN DIVIDE BY 2.
SO WE'LL ADD TO BOTH SIDES AND THEN DIVIDE BY 2.
00:53

SO SINE THETA MUST EQUAL 1/2.
SO SINE THETA MUST EQUAL 1/2.
00:56

FIRST WE WANT TO FIND ALL THE ANGLES
FIRST WE WANT TO FIND ALL THE ANGLES
00:58

THAT HAVE A SINE FUNCTION VALUE OF 1/2
THAT HAVE A SINE FUNCTION VALUE OF 1/2
01:01

ON THE INTERVAL FROM ZERO TO 2PI.
ON THE INTERVAL FROM ZERO TO 2PI.
01:04

NOW, WE COULD USE REFERENCE TRIANGLES,
NOW, WE COULD USE REFERENCE TRIANGLES,
01:06

BUT IN THIS CASE I'M GOING TO GO AHEAD
BUT IN THIS CASE I'M GOING TO GO AHEAD
01:07

AND USE THE UNIT CIRCLE.
AND USE THE UNIT CIRCLE.
01:09

REMEMBER, THE Y COORDINATE ON THE UNIT CIRCLE
REMEMBER, THE Y COORDINATE ON THE UNIT CIRCLE
01:11

IS EQUAL TO SINE THETA.
IS EQUAL TO SINE THETA.
01:14

SO FIRST, IF WE KNOW THAT THE Y COORDINATE WILL BE +1/2
SO FIRST, IF WE KNOW THAT THE Y COORDINATE WILL BE +1/2
01:19

OUR ANGLES MUST BE IN EITHER THE FIRST OR SECOND QUADRANT.
OUR ANGLES MUST BE IN EITHER THE FIRST OR SECOND QUADRANT.
01:23

SO WE'RE LOOKING FOR A Y COORDINATE OF 1/2,
SO WE'RE LOOKING FOR A Y COORDINATE OF 1/2,
01:25

AND RIGHT AWAY WE SEE ONE HERE,
AND RIGHT AWAY WE SEE ONE HERE,
01:27

30 DEGREES OR PI/6 RADIANS.
30 DEGREES OR PI/6 RADIANS.
01:34

NOTICE WITH A 30 DEGREE REFERENCE ANGLE
NOTICE WITH A 30 DEGREE REFERENCE ANGLE
01:36

IN THE SECOND QUADRANT
IN THE SECOND QUADRANT
01:37

THAT PRODUCES AN ANGLE OF 150 DEGREES OR 5PI/6 RADIANS.
THAT PRODUCES AN ANGLE OF 150 DEGREES OR 5PI/6 RADIANS.
01:42

AND, AGAIN, THE Y COORDINATE IS EQUAL TO 1/2,
AND, AGAIN, THE Y COORDINATE IS EQUAL TO 1/2,
01:45

THEREFORE, SINE THETA IS EQUAL TO 1/2.
THEREFORE, SINE THETA IS EQUAL TO 1/2.
01:48

SO WE HAVE TWO ANGLES ON THE INTERVAL FROM ZERO TO 2PI
SO WE HAVE TWO ANGLES ON THE INTERVAL FROM ZERO TO 2PI
01:52

THAT PRODUCE A SINE FUNCTION VALUE OF 1/2.
THAT PRODUCE A SINE FUNCTION VALUE OF 1/2.
01:57

SO THAT TAKES CARE OF THE INTERVAL FROM ZERO TO 2PI.
SO THAT TAKES CARE OF THE INTERVAL FROM ZERO TO 2PI.
02:00

NOW, IF WE WANT TO FIND ALL RADIAN SOLUTIONS,
NOW, IF WE WANT TO FIND ALL RADIAN SOLUTIONS,
02:03

REMEMBER THAT THE PERIOD FOR THE SINE FUNCTION
REMEMBER THAT THE PERIOD FOR THE SINE FUNCTION
02:07

IS 2PI RADIANS.
IS 2PI RADIANS.
02:08

SO IF WE TAKE THIS ANGLE
SO IF WE TAKE THIS ANGLE
02:10

AND ADD ANY MULTIPLE OF 2PI RADIANS
AND ADD ANY MULTIPLE OF 2PI RADIANS
02:13

WE'LL HAVE ANOTHER COTERMINAL ANGLE WITH THIS ANGLE
WE'LL HAVE ANOTHER COTERMINAL ANGLE WITH THIS ANGLE
02:16

AND, THEREFORE, IT'LL HAVE THE SAME SINE FUNCTION VALUE.
AND, THEREFORE, IT'LL HAVE THE SAME SINE FUNCTION VALUE.
02:20

SO THETA COULD EQUAL PI/6 RADIANS PLUS--
SO THETA COULD EQUAL PI/6 RADIANS PLUS--
02:26

WE'LL SAY 2PI K WHERE K IS SUM INTEGER.
WE'LL SAY 2PI K WHERE K IS SUM INTEGER.
02:32

SO THIS WILL PRODUCE SUM MULTIPLE OF 2PI RADIANS.
SO THIS WILL PRODUCE SUM MULTIPLE OF 2PI RADIANS.
02:35

OR THETA COULD ALSO BE 5PI/6 RADIANS + 2PI K.
OR THETA COULD ALSO BE 5PI/6 RADIANS + 2PI K.
02:43

AND THAT WOULD PRODUCE ANY COTERMINAL ANGLE
AND THAT WOULD PRODUCE ANY COTERMINAL ANGLE
02:45

WITH THIS TERMINAL SIDE
WITH THIS TERMINAL SIDE
02:46

PRODUCING A SINE FUNCTION VALUE OF 1/2 AS WELL.
PRODUCING A SINE FUNCTION VALUE OF 1/2 AS WELL.
02:50

LET'S LOOK AT IT GRAPHICALLY NOW.
LET'S LOOK AT IT GRAPHICALLY NOW.
02:52

HERE WE HAVE THE GRAPH OF Y = SINE THETA AND THEN Y = 1/2.
HERE WE HAVE THE GRAPH OF Y = SINE THETA AND THEN Y = 1/2.
02:57

THESE INTERSECTION POINTS REPRESENT SOLUTIONS.
THESE INTERSECTION POINTS REPRESENT SOLUTIONS.
03:01

SO THE FIRST TWO SOLUTIONS WE FOUND FROM ZERO TO 2PI
SO THE FIRST TWO SOLUTIONS WE FOUND FROM ZERO TO 2PI
03:05

WERE HERE AND HERE.
WERE HERE AND HERE.
03:08

THESE OTHER POINTS OF INTERSECTIONS
THESE OTHER POINTS OF INTERSECTIONS
03:10

WOULD BE OBTAINED BY TAKING ONE OF THESE ANGLES
WOULD BE OBTAINED BY TAKING ONE OF THESE ANGLES
03:12

AND ADDING A MULTIPLE OF 2PI RADIANS.
AND ADDING A MULTIPLE OF 2PI RADIANS.
03:16

THE DISTANCE FROM THIS POINT TO THIS POINT IS 2PI RADIANS
THE DISTANCE FROM THIS POINT TO THIS POINT IS 2PI RADIANS
03:21

AND THE DISTANCE FROM THIS POINT TO THIS POINT
AND THE DISTANCE FROM THIS POINT TO THIS POINT
03:23

IS ANOTHER 2PI RADIANS.
IS ANOTHER 2PI RADIANS.
03:25

SO YOU KIND OF SEE
SO YOU KIND OF SEE
03:26

WHAT'S HAPPENING HERE GRAPHICALLY AS WELL.
WHAT'S HAPPENING HERE GRAPHICALLY AS WELL.
03:28

HERE WE HAVE ANOTHER EQUATION,
HERE WE HAVE ANOTHER EQUATION,
03:29

LET'S GO AHEAD AND SOLVE THIS FOR COSINE THETA.
LET'S GO AHEAD AND SOLVE THIS FOR COSINE THETA.
03:32

SO WE'LL SUBTRACT THE SQUARE ROOT OF 2 ON BOTH SIDES,
SO WE'LL SUBTRACT THE SQUARE ROOT OF 2 ON BOTH SIDES,
03:36

THEN DIVIDE BY 2.
THEN DIVIDE BY 2.
03:41

COSINE THETA MUST EQUAL -SQUARE ROOT 2/2.
COSINE THETA MUST EQUAL -SQUARE ROOT 2/2.
03:45

WELL, AGAIN, USING A UNIT CIRCLE,
WELL, AGAIN, USING A UNIT CIRCLE,
03:48

COSINE THETA IS EQUAL TO X.
COSINE THETA IS EQUAL TO X.
03:50

SO NOW WE'LL LOOK FOR AN X COORDINATE
SO NOW WE'LL LOOK FOR AN X COORDINATE
03:52

OF -SQUARE ROOT 2/2.
OF -SQUARE ROOT 2/2.
03:54

AND THAT'S GOING TO LIMIT US
AND THAT'S GOING TO LIMIT US
03:55

TO THE SECOND AND THIRD QUADRANT
TO THE SECOND AND THIRD QUADRANT
03:58

SINCE THE X COORDINATE IS GOING TO BE NEGATIVE.
SINCE THE X COORDINATE IS GOING TO BE NEGATIVE.
04:01

SO LOOKING IN THE SECOND QUADRANT,
SO LOOKING IN THE SECOND QUADRANT,
04:02

HERE'S AN X COORDINATE OF -SQUARE ROOT 2/2.
HERE'S AN X COORDINATE OF -SQUARE ROOT 2/2.
04:06

THE ANGLE IS 135 DEGREES OR 3PI/4 RADIANS.
THE ANGLE IS 135 DEGREES OR 3PI/4 RADIANS.
04:12

AND IN THE THIRD QUADRANT,
AND IN THE THIRD QUADRANT,
04:15

HERE'S THE COSINE VALUE OF - SQUARE ROOT 2/2,
HERE'S THE COSINE VALUE OF - SQUARE ROOT 2/2,
04:17

WHICH IS 5PI/4 RADIANS.
WHICH IS 5PI/4 RADIANS.
04:21

AND, AGAIN, TO FIND ALL RADIAN SOLUTIONS
AND, AGAIN, TO FIND ALL RADIAN SOLUTIONS
04:23

WE'LL JUST HAVE TO ADD MULTIPLES OF 2PI RADIANS
WE'LL JUST HAVE TO ADD MULTIPLES OF 2PI RADIANS
04:25

TO BOTH OF THESE.
TO BOTH OF THESE.
04:33

SO TO SOLVE THESE TYPES OF EQUATIONS
SO TO SOLVE THESE TYPES OF EQUATIONS
04:35

WITH SINE AND COSINE THE UNIT CIRCLE COMES IN VERY HANDY.
WITH SINE AND COSINE THE UNIT CIRCLE COMES IN VERY HANDY.
04:38

LET'S TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
LET'S TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
04:41

HERE'S THE GRAPH OF Y = COSINE THETA
HERE'S THE GRAPH OF Y = COSINE THETA
04:44

AND WE HAVE THE GRAPH OF Y = - SQUARE ROOT 2/2.
AND WE HAVE THE GRAPH OF Y = - SQUARE ROOT 2/2.
04:49

THE SOLUTIONS THAT WE FOUND ON THE INTERVAL FROM ZERO TO 2PI
THE SOLUTIONS THAT WE FOUND ON THE INTERVAL FROM ZERO TO 2PI
04:53

WERE HERE AND HERE.
WERE HERE AND HERE.
04:56

AND THESE OTHER SOLUTIONS WE SEE
AND THESE OTHER SOLUTIONS WE SEE
04:57

WOULD BE THESE VALUES PLUS OR MINUS MULTIPLES OF 2PI.
WOULD BE THESE VALUES PLUS OR MINUS MULTIPLES OF 2PI.
05:04

OKAY, LET'S TAKE A LOOK AT AN EQUATION NOW
OKAY, LET'S TAKE A LOOK AT AN EQUATION NOW
05:06

THAT INVOLVES TANGENT THETA.
THAT INVOLVES TANGENT THETA.
05:07

AGAIN, WE'LL FIRST SOLVE THIS FOR TAN THETA
AGAIN, WE'LL FIRST SOLVE THIS FOR TAN THETA
05:10

BY ADDING 1 TO BOTH SIDES, DIVIDING BY SQUARE ROOT 3.
BY ADDING 1 TO BOTH SIDES, DIVIDING BY SQUARE ROOT 3.
05:16

SO WE HAVE TAN THETA = 1/SQUARE ROOT 3.
SO WE HAVE TAN THETA = 1/SQUARE ROOT 3.
05:21

NOW, THIS RATIO SHOULD REMIND YOU
NOW, THIS RATIO SHOULD REMIND YOU
05:23

OF 30, 60, 90 RIGHT TRIANGLE.
OF 30, 60, 90 RIGHT TRIANGLE.
05:26

IF YOU TAKE A LOOK AT A 30 DEGREE ANGLE,
IF YOU TAKE A LOOK AT A 30 DEGREE ANGLE,
05:28

THE TANGENT RATIO
THE TANGENT RATIO
05:29

OR THE OPPOSITE SIDE OVER THE ADJACENT SIDE
OR THE OPPOSITE SIDE OVER THE ADJACENT SIDE
05:31

IS 1/SQUARE ROOT 3.
IS 1/SQUARE ROOT 3.
05:34

SO THAT GIVES US OUR FIRST MEASURE FOR ANGLE THETA
SO THAT GIVES US OUR FIRST MEASURE FOR ANGLE THETA
05:37

AS 30 DEGREES OR PI/6 RADIANS.
AS 30 DEGREES OR PI/6 RADIANS.
05:41

LET'S ALSO SKETCH THIS ON THE COORDINATE PLANE.
LET'S ALSO SKETCH THIS ON THE COORDINATE PLANE.
05:44

NOW, REMEMBER THAT TANGENT THETA IS ALSO POSITIVE
NOW, REMEMBER THAT TANGENT THETA IS ALSO POSITIVE
05:47

IN A THIRD QUADRANT
IN A THIRD QUADRANT
05:48

WHERE BOTH THE X AND THE Y COORDINATES ARE NEGATIVE.
WHERE BOTH THE X AND THE Y COORDINATES ARE NEGATIVE.
05:51

SO IF WE SKETCH A 30 DEGREE REFERENCE ANGLE
SO IF WE SKETCH A 30 DEGREE REFERENCE ANGLE
05:54

IN THE THIRD QUADRANT
IN THE THIRD QUADRANT
05:55

THIS ANGLE WOULD ALSO HAVE A TANGENT FUNCTION VALUE
THIS ANGLE WOULD ALSO HAVE A TANGENT FUNCTION VALUE
05:59

OF 1/SQUARE ROOT 3.
OF 1/SQUARE ROOT 3.
06:02

AND THIS ANGLE WOULD BE 180 + 30 OR 210 DEGREES,
AND THIS ANGLE WOULD BE 180 + 30 OR 210 DEGREES,
06:07

WHICH IS EQUAL TO 7PI/6 RADIANS.
WHICH IS EQUAL TO 7PI/6 RADIANS.
06:11

NOW, FOR ALL RADIAN SOLUTIONS,
NOW, FOR ALL RADIAN SOLUTIONS,
06:13

REMEMBER THE PERIOD FOR TANGENT THETA
REMEMBER THE PERIOD FOR TANGENT THETA
06:15

IS PI RADIANS OR 180 DEGREES.
IS PI RADIANS OR 180 DEGREES.
06:19

SO FOR THIS PROBLEM WE'RE ONLY GOING TO HAVE
SO FOR THIS PROBLEM WE'RE ONLY GOING TO HAVE
06:21

ONE EQUATION FOR THETA AND IT WILL EQUAL--
ONE EQUATION FOR THETA AND IT WILL EQUAL--
06:24

WE CAN USE EITHER OF THESE ANGLES,
WE CAN USE EITHER OF THESE ANGLES,
06:25

BUT LET'S USE PI/6 PLUS ANY MULTIPLE OF PI RADIANS.
BUT LET'S USE PI/6 PLUS ANY MULTIPLE OF PI RADIANS.
06:31

SO WE'LL GET THIS SECOND ANGLE AND ANY OTHER COTERMINAL ANGLE
SO WE'LL GET THIS SECOND ANGLE AND ANY OTHER COTERMINAL ANGLE
06:37

BY ADDING MULTIPLES OF PI TO PI/6 RADIANS.
BY ADDING MULTIPLES OF PI TO PI/6 RADIANS.
06:42

SO WE DON'T NEED A SECOND EQUATION HERE
SO WE DON'T NEED A SECOND EQUATION HERE
06:45

FOR ALREADY IN MEASURE DUE TO THE PERIOD BEING PI
FOR ALREADY IN MEASURE DUE TO THE PERIOD BEING PI
06:47

INSTEAD OF 2PI.
INSTEAD OF 2PI.
06:49

LET'S LOOK AT IT GRAPHICALLY.
LET'S LOOK AT IT GRAPHICALLY.
06:51

HERE ARE THE TWO SOLUTIONS WE LISTED FROM ZERO TO 2PI,
HERE ARE THE TWO SOLUTIONS WE LISTED FROM ZERO TO 2PI,
06:54

BUT YOU CAN SEE ALL OTHER SOLUTIONS
BUT YOU CAN SEE ALL OTHER SOLUTIONS
06:55

WILL BE MULTIPLES OF PI FROM EITHER OF THESE.
WILL BE MULTIPLES OF PI FROM EITHER OF THESE.
06:58

WE CHOSE PI/6 FOR OUR EQUATION.
WE CHOSE PI/6 FOR OUR EQUATION.
07:01

LET'S TAKE A LOOK AT ANOTHER EQUATION.
LET'S TAKE A LOOK AT ANOTHER EQUATION.
07:03

4 COSINE THETA - 6 = COSINE THETA.
4 COSINE THETA - 6 = COSINE THETA.
07:06

WELL, IN ORDER TO SOLVE THIS FOR COSINE THETA
WELL, IN ORDER TO SOLVE THIS FOR COSINE THETA
07:08

WE MUST GET COSINE THETA ON THE SAME SIDE.
WE MUST GET COSINE THETA ON THE SAME SIDE.
07:11

SO WE'LL SUBTRACT COSINE THETA, ADD 6 TO BOTH SIDES,
SO WE'LL SUBTRACT COSINE THETA, ADD 6 TO BOTH SIDES,
07:21

THEN DIVIDE BY 3, COSINE THETA = 2.
THEN DIVIDE BY 3, COSINE THETA = 2.
07:28

REMEMBER THE RANGE FOR COSINE THETA
REMEMBER THE RANGE FOR COSINE THETA
07:30

IS ACTUALLY THE CLOSED INTERVAL FROM -1 TO +1.
IS ACTUALLY THE CLOSED INTERVAL FROM -1 TO +1.
07:35

SO COSINE THETA WILL NEVER EQUAL 2,
SO COSINE THETA WILL NEVER EQUAL 2,
07:37

THEREFORE, FOR THIS EQUATION WE HAVE NO SOLUTION.
THEREFORE, FOR THIS EQUATION WE HAVE NO SOLUTION.
07:43

LET'S GO AND TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
LET'S GO AND TAKE A LOOK AT THIS ONE GRAPHICALLY AS WELL.
07:46

HERE WE HAVE THE GRAPH OF Y = 2
HERE WE HAVE THE GRAPH OF Y = 2
07:48

AND THERE'S Y = COSINE THETA.
AND THERE'S Y = COSINE THETA.
07:52

SINCE THEY DON'T INTERSECT
SINCE THEY DON'T INTERSECT
07:54

THIS VERIFIES OUR ANSWER OF NO SOLUTION.
THIS VERIFIES OUR ANSWER OF NO SOLUTION.
07:57

OKAY. I HOPE YOU HAVE FOUND THIS VIDEO HELPFUL.
OKAY. I HOPE YOU HAVE FOUND THIS VIDEO HELPFUL.
07:59

THANK YOU FOR WATCHING
THANK YOU FOR WATCHING
08:00

AND REMEMBER THIS IS JUST THE FIRST OF SEVERAL VIDEOS
AND REMEMBER THIS IS JUST THE FIRST OF SEVERAL VIDEOS
08:03

ON SOLVING TRIG EQUATIONS.
ON SOLVING TRIG EQUATIONS.

All noun

series /ˈsirēz/ word	several events, objects, or people of similar or related kind coming one after another

Solving Trigonometric Equations I

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Mathispower4u

Video Language:

English

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English (en)

Accent:

English (US)

Speech Time:

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8:00 / 8:19

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Intro:

- 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..