- WELCOME TO AN INTRODUCTION
TO INVERSE TRIGNOMETRIC FUNCTIONS.
THE GOALS OF THE VIDEO ARE TO USE INVERSE TRIG FUNCTIONS
TO FIND AN ANGLE OF A GIVEN TRIGNOMETRIC FUNCTION VALUE,
AND ALSO TO USE INVERSE TRIGOMETRIC FUNCTIONS
TO SOLVE A RIGHT TRIANGLE.
REMEMBER, YOU CAN THINK OF A FUNCTION
AND ITS INVERSE FUNCTION
AS TWO FUNCTIONS THAT UNDO EACH OTHER.
HERE'S AN ILLUSTRATION OF THIS.
LET'S SEE WE HAVE THIS FUNCTION F,
AND X IS THE INPUT AND Y IS THE OUTPUT.
WELL, THE OUTPUT OF F BECOMES THE INPUT OF F INVERSE
AND F INVERSE OUTPUTS X.
SO WHAT WE SEE HERE IS WHATEVER WE START WITH
WE END UP WITH THE SAME THING
IF THESE TWO FUNCTIONS ARE INVERSES OF ONE ANOTHER.
HERE'S A WAY TO LOOK AT IT FROM A TRIG STANDPOINT.
IF I LET F OF X = SIN X, AND LET THE INVERSE BE INVERSE SIN.
LET'S SAY, FOR EXAMPLE, I INPUT 30 DEGREES.
WELL, THE SIN OF 30 DEGREES IS EQUAL TO 1/2.
AND IF I PUT THE SIN RATIO INTO THE INVERSE FUNCTION
IT WILL RETURN THE ANGLE AGAIN.
SO THE FINAL RESULT WILL BE 30 DEGREES AGAIN.
INVERSE TRIGOMETRIC FUNCTIONS ARE WRITTEN AS WE SEE HERE.
Y = INVERSE SIN X.
BUT WE CAN ALSO WRITE THIS AS ARC SIN X.
AND INVERSE COSINE X CAN BE WRITTEN ARC COSINE X.
AND INVERSE TANGENT X CAN BE WRITTEN AS ARC TANGENT X.
THE THING TO REMEMBER IS THAT
FOR THESE INVERSE TRIGOMETRIC FUNCTIONS,
THE INPUT IS A TRIG FUNCTION VALUE
AND THE OUTPUT IS AN ANGLE.
AND WE CAN USE INVERSE TRIG FUNCTIONS
TO SOLVE THE FOLLOWING TYPES OF PROBLEMS.
USE THE CALCULATOR TO FIND THE ANGLE, THETA
ON THE INTERVAL FROM ZERO TO 90 DEGREES
THAT SATISFIES THE EQUATION.
SO WE KNOW THAT SIN THETA IS EQUAL TO 0.7523,
AND OUR GOAL IS TO FIND THE ANGLE
THAT WOULD GIVE THIS SIN VALUE.
SO WHAT WE CAN DO IS TAKE THIS EQUATION
AND TAKE THE INVERSE SIN OF BOTH SIDES.
NOW, REMEMBER THAT THESE TWO FUNCTIONS
ARE INVERSES OF ONE ANOTHER.
SO THEY WILL UNDO EACH OTHER.
AND THE RESULT OF THIS WOULD JUST BE THETA ON THE LEFT SIDE.
AND ON THE RIGHT SIDE THIS --
AND THIS IS NOT ONE OF THOSE RATIOS
THAT WE ARE USED TO SEEING
IN REGARDS TO ONE OF THE 30, 45, OR 60 REFERENCE ANGLES.
SO WE WILL HAVE TO USE OUR GRAPHING CALCULATOR.
SO LET'S GO AHEAD AND GET THAT OUT.
AND NOW WE WILL USE THE INVERSE SIN KEY,
WHICH IS HERE, RIGHT ABOVE THE SIN KEY.
SO IF YOU PRESS SECOND SIN,
IT BRINGS UP THE INVERSE SIN FUNCTION.
THEN WE JUST TYPE IN 0.7523,
AND THIS WILL RETURN THE ANGLE WITHOUT HAS THAT SIN VALUE.
SO IT'S ROUGHLY PLOT 48.8 DEGREES.
LET'S GO AHEAD AND TRY ANOTHER ONE.
HERE WE HAVE 10 THETA = 3.54.
SO, AGAIN, IN ORDER TO FIND THETA
WE CAN TAKE THE ARC TANGENT
OR THE INVERSE TANGENT OF BOTH SIDES.
AND, AGAIN, THESE TWO UNDO EACH OTHER
AND THE RESULT IS JUST THETA.
AND NOW WE CAN FIND THIS ON OUR GRAPHING CALCULATOR.
SO RIGHT ABOVE THE TANGENT WE HAVE THE ARC TANGENT
OR THE INVERSE TANGENT KEY.
PRESS SECOND TANGENT.
TYPE IN 3.54, AND THIS RETURNS THE ANGEL ROUGHLY 74.2 DEGREES.
THERE'S A LOT MORE TO SAY ABOUT INVERSE TRIGOMETRIC FUNCTIONS.
HOWEVER, FOR RIGHT NOW WE'RE JUST GOING TO USE THEM
TO FIND ANGLES GIVEN A TRIG FUNCTIONS VALUE,
AND ALSO TO SOLVE RIGHT TRIANGLES.
SO HERE'S ANOTHER EXAMPLE.
WE WANT TO SOLVE THE GIVEN RIGHT TRIANGLE
IF A = 44.3 CENTIMETERS, AND B = 55.9 CENTIMETERS.
LET'S CALL THIS ANGLE A, ANGLE B, ANGLE C.
SO SIDE A ACROSS FROM ANGLE A, 44.3 CENTIMETERS.
B, 55.9 CENTIMETERS.
WE NEED TO FIND THE HYPOTENUSE AND THE INTERIOR ANGLES
OF THIS TRIANGLE.
WE CAN FIND THE HYPOTENUSE BY USING THE PYTHAGOREAN THEOREM.
C SQUARED = A SQUARED + B SQUARED.
SO C WILL EQUAL THE SQUARE ROOT OF THIS.
LET'S GO AHEAD AND GO TO OUR CALCULATOR.
AND C IS APPROXIMATELY 71.3 CENTIMETERS.
NOW, WE NEED TO FIND ANGLE A AND ANGLE B AS WELL.
AND SO WHAT WE CAN DO NOW IS-- SO IF WE LOOK AT ANGLE A,
WE CAN PICK ANY OF THE THREE GEOMETRIC FUNCTIONS.
LET'S LOOK AT TANGENT.
THE TANGENT OF ANGLE A
MUST EQUAL THE RATIO OF THE OPPOSITE SIDE
TO THE ADJACENT SIDE.
SO 44.3 DIVIDED BY 55.9.
AND WE CAN FIND ANGLE A
IF WE TAKE THE ARC TANGENT OF BOTH SIDES.
THE LEFT SIDE WILL GIVE US ANGLE A.
AND WE CAN FIND THIS VALUE ON THE GRAPHING CALCULATOR.
SO SECOND TANGENT.
AND WE CAN SEE OUR ANGLE IS ROUGHLY 38.4 DEGREES.
NOW, WE KNOW THAT THESE TWO ANGLES, A AND B
MUST BE COMPLEMENTARY.
SO IF 90 - 38.4 WOULD GIVE US 51.6 DEGREES FOR ANGLE B,
AND WE HAVE SOLVED THE RIGHT TRIANGLE.
LET'S TAKE A LOOK AT ONE MORE.
FIND THE ANGLES IN THE THREE, FOUR, FIVE RIGHT TRIANGLE.
REMEMBER THREE, FOUR, FIVE
IS ONE OF THOSE SPECIAL RIGHT TRIANGLES.
BUT WE NEVER TALK ABOUT THAT THE ANGLES OF THAT TRIANGLE ARE.
AND NOW WE'RE GOING TO FIGURE IT OUT.
SO LET'S GO AHEAD AND JUST CALL THIS ANGLE A, ANGLE B,
AND ANGLE C.
WE NEED TO FIND ANGLE A AND ANGLE B.
SO IF I TAKE LOOK AT ANGLE A--
I'M GOING TO GO AHEAD AND USE SIN THIS TIME.
SO IF THE SIN OF ANGLE A MUST EQUAL THE OPPOSITE
OR THE HYPOTENUSE.
SO SIN OF A IS EQUAL TO 3/5.
SO, AGAIN, IN ORDER TO FIND ANGLE A,
I CAN TAKE THE INVERSE SIN OF BOTH SIDES.
SO THE LEFT SIDE WILL HAVE ANGLE A,
AND THE RIGHT SIDE WILL GO TO OUR CALCULATOR.
WE WANT THE INVERSE SIN OF 3/5.
SO SECOND SIN 3/5.
IT LOOKS LIKE IT'S ROUGHLY 36.9 DEGREES.
AND, AGAIN, THESE TWO ARE COMPLIMENTARY.
SO 90 - 36.9 WILL GIVE US 53.1 DEGREES FOR ANGLE B.
AND THAT TRIANGLE IS SOLVED.
AGAIN, WE WILL DISCUSS MUCH MORE
ABOUT INVERSE TRIGOMETRIC FUNCTIONS LATER,
BUT FOR RIGHT NOW WE CAN USE THIS INFORMATION
TO SOLVE RIGHT TRIANGLES.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
HAVE A GOOD DAY.