- NOW THAT WE'VE LEARNED ABOUT INVERSE TRIG FUNCTIONS
WE CAN NOW SOLVE RIGHT TRIANGLES THAT WE COULDN'T SOLVE BEFORE.
LET'S TAKE A LOOK AT AN EXAMPLE.
NOW REMEMBER TO SOLVE A RIGHT TRIANGLE
MEANS TO FIND THE LENGTHS OF ALL THE SIDES
AND THE MEASURE OF ALL OF THE ANGLES.
NOTICE WE'RE GIVEN THE LENGTHS OF TWO OF THE SIDES
AS WELL AS WE KNOW THE MEASURE OF ANGLE "C" IS 90 DEGREES.
SO WE NEED TO FIND THE MEASURE OF ANGLE "A",
THE MEASURE OF ANGLE "B" AND THE LENGTH OF SIDE "C".
LET'S START BY DETERMINING THE LENGTH OF SIDE "C".
AND SINCE WE HAVE A RIGHT TRIANGLE
WE CAN USE THE PYTHAGOREAN THEOREM.
SO WE KNOW "C" SQUARED
MUST BE EQUAL TO "A" SQUARED + "B" SQUARED
WHERE "C" IS THE UNKNOWN.
SO WE HAVE "C" SQUARED = "A" SQUARED.
WELL, "A" IS THE SIDE THAT'S OPPOSITE ANGLE "A"
SO "A" IS EQUAL TO 6 METERS.
SO WE'LL HAVE 6 SQUARED + "B" SQUARED WHERE "B" IS 12 METERS.
SO WE'LL HAVE 12 SQUARED.
SO WE HAVE "C" SQUARED EQUALS-- IT WOULD BE 36 + 144.
SO WE HAVE "C" SQUARED EQUALS-- THIS WOULD BE 180.
AND NOW TO FIND "C"
WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION.
WE ONLY HAVE TO TAKE THE PRINCIPAL SQUARE ROOT
BECAUSE WE KNOW "C" HAS TO BE POSITIVE.
SO WE HAVE "C" EQUALS THE SQUARE ROOT OF 180.
BUT THIS CAN BE SIMPLIFIED.
180 IS EQUAL TO 9 x 20 WHERE 9 IS PERFECT SQUARE.
AND 20 IS EQUAL TO 4 x 5 WHERE FOUR IS A PERFECT SQUARE.
NOTICE THAT 9 x 4 IS 36 ALSO A PERFECT SQUARE.
WITH MEANS YOU CAN WRITE SQUARE ROOT OF 180 AS 36 x 5.
AND THE SQUARE ROOT OF 36 IS EQUAL TO 6.
SO THE EXACT VALUE OF "C" WOULD BE 6 SQUARE ROOT 5,
OF COURSE, THIS WOULD BE METERS.
LET'S ALSO FIND OUR DECIMAL APPROXIMATION FOR THIS.
6, SECOND "X" SQUARED BRINGS UP THE SQUARE ROOT, 5 AND ENTER.
AND TO VERIFY THAT THIS IS CORRECT,
LET'S ALSO COMPARE THIS TO SQUARE ROOT 180.
THIS VERIFIES THAT WE DID SIMPLIFY
OUR SQUARE ROOT CORRECTLY.
SO "C" IS APPROXIMATELY 13.42 METERS.
AND NOW LET'S FIND THE MEASURE OF ANGLE "A".
NOW, NOTICE THAT WE CAN USE ANY OF THE THREE SIDES
OF THE RIGHT TRIANGLE DEPENDING ON WHAT TRIG FUNCTION
WE WANT TO USE.
IF WE WANT TO USE THE TANGENT OF ANGLE "A"
WHERE THE TANGENT FUNCTION IS EQUAL TO THE RATIO
OF THE OPPOSITE SIDE TO THE ADJACENT SIDE
WE COULD SAY TANGENT "A" IS EQUAL TO 6/12.
IF WE DID WANT TO USE, LET'S SAY, THE SINE FUNCTION.
THE SINE FUNCTION VALUE IS THE RATIO
OF THE OPPOSITE SIDE TO THE HYPOTENUSE.
WE COULD WRITE SINE "A" EQUALS 6 DIVIDED BY 6,
SQUARE ROOT 5 WE PROBABLY WOULD NOT WANT TO USE
YOUR ROUNDED VALUE
BECAUSE THAT WOULD GIVE US MORE OF AN ERROR
IN DETERMINING THE MEASURE OF ANGLE "A".
BUT FOR THIS EXAMPLE LET'S GO AHEAD
AND USE THE TANGENT FUNCTION
WHICH MEANS THE TANGENT OF ANGLE "A" MUST BE EQUAL TO THE RATIO
OF THE OPPOSITE SIDE TO THE ADJACENT SIDE
WHICH WOULD BE 6/12,
OF COURSE, THIS SIMPLIFIES TO 1/2.
SO WE HAVE TANGENT "A" IS EQUAL TO 1/2.
NOW WE CAN USE INVERSE TRIG FUNCTIONS
TO FIND THE VALUE OF "A".
IF WE WANT TO SOLVE THIS EQUATION FOR "A"
WE CAN TAKE THE INVERSE TANGENT OR ARC TANGENT OF BOTH SIDES
OF THE EQUATION.
[ CLICKING SOUNDS ]
- BECAUSE ARC TANGENT AND TANGENT OR INVERSE FUNCTIONS
THEY UNDO EACH OTHER AND THEREFORE ARC TANGENT,
TANGENT "A" IS JUST EQUAL TO ANGLE "A".
AND ON THE RIGHT SIDE OF THE EQUATION
WE HAVE ARC TANGENT 1/2 OR INVERSE TANGENT 1/2
WHERE ONE HALF IS A TANGENT FUNCTION VALUE
AND THIS EXPRESSION IS EQUAL TO THE ANGLE
THAT HAS A TANGENT FUNCTION VALUE OF 1/2.
SO TO FIND THIS ANGLE WE'LL HAVE TO USE A CALCULATOR
BECAUSE THIS TRIG FUNCTION VALUE DOES NOT RESULT
IN A NICE REFERENCE ANGLE.
SO NOW WE'LL GO TO THE CALCULATOR.
WE WANT OUR ANGLE TO BE IN DEGREES.
SO LET'S GO AHEAD AND CHECK OUR MODE.
SO I'LL PRESS THE MODE KEY.
WE'LL CHANGE THE THIRD ROW TO DEGREE.
HIGHLIGHT IT AND PRESS ENTER.
BACK TO THE HOME SCREEN.
AND NOW WE'RE GOING TO PRESS SECOND TANGENT
FOR INVERSE TANGENT OR ARC TANGENT
AND THEN ENTER 1/2 AND ENTER.
SO ANGLE "A" IS APPROXIMATELY--
LET'S FIND THE TWO DECIMAL PLACES.
LET'S SAY IT'S APPROXIMATELY 26.57 DEGREES.
[ CLICKING SOUNDS ]
- NOW BEFORE WE FIND THE MEASURE OF ANGLE "B"
LET'S GO AND FIND THE MEASURE OF ANGLE "A" AGAIN
USING A DIFFERENT TRIG FUNCTION.
LET'S SAY WE WANTED TO USE THE SINE FUNCTION VALUE INSTEAD.
REMEMBER THE SINE FUNCTION IS EQUAL TO THE RATIO
OF THE OPPOSITE SIDE OF THE HYPOTENUSE.
SO, AGAIN, IF WE REFERENCE ANGLE "A"
WE CAN SAY SINE "A" IS EQUAL TO THE OPPOSITE SIDE
WHICH IS 6 DIVIDED BY THE HYPOTENUSE
WHICH IS 6 SQUARE ROOT 5.
WELL, AGAIN, NOTICE HOW THIS SIMPLIFIES NICELY TO 1
DIVIDED BY SQUARE ROOT 5.
SO WE HAVE SIGN "A" EQUALS--
THIS WOULD BE, AGAIN, 1 DIVIDED BY THE SQUARE ROOT OF 5.
AND NOW WE'LL TAKE THE INVERSE SIGN OR ARC SIGN OF BOTH SIDES
OF THE EQUATION TO SOLVE THIS FOR "A".
AND, AGAIN, ON THE LEFT SIDE
IF WE TAKE THE INVERSE SIGN OF SINE "A" THE TWO FUNCTIONS
UNDO EACH OTHER BECAUSE THEY'RE INVERSES.
SO WE HAVE ANGLE "A" IS EQUAL TO INVERSE SINE
OF ONE DIVIDED BY SQUARE ROOT 5.
NOW WE'LL GO BACK TO THE CALCULATOR
TO MAKE SURE WE GET THE SAME ANGLE
THAT WE DID WHEN WE USED THE TANGENT
AND INVERSE TANGENT FUNCTION.
SO NOW WE'RE GOING TO PRESS SECOND SINE FOR INVERSE SIGN
FOR INVERSE SINE OR ARC SINE.
AND IT WAS 1 DIVIDED BY SQUARE ROOT 5.
NOTICE HOW, OF COURSE, ANGLE "A" IS STILL THE SAME.
IT'S APPROXIMATELY 26.57
AND THIS WAS DEGREES.
AND, OF COURSE, BACK OVER HERE
WE SHOULD CHANGE THIS EQUAL SIGN TO AN APPROXIMATION SYMBOL.
NOW TO FIND ANGLE "B" WE COULD USE ANOTHER TRIG EQUATION
BUT REMEMBER FOR A RIGHT TRIANGLE THE TWO ACUTE ANGLES
MUST BE COMPLEMENTARY OR HAVE A SUM OF 90 DEGREES.
SO WE CAN SAY THAT ANGLE "B" IS GOING TO BE APPROXIMATELY
EQUAL TO 90 DEGREES - THE MEASURE OF ANGLE "A"
WHICH IS 26.57 DEGREES WHICH MEANS ANGLE "B"
WOULD BE APPROXIMATELY 63.43 DEGREES.
AND NOW WE'VE SOLVED THE TRIANGLE.
WE WERE GIVEN TWO SIDES AND AN ANGLE
AND WE FOUND THE MEASURE OF ANGLE "A"
AS WELL AS THE MEASURE OF ANGLE "B" AND THE LENGTH OF SIDE "C".
I HOPE YOU FOUND THIS HELPFUL.