- WELCOME TO A VIDEO
THAT INTRODUCES THE TRIGONOMETRIC FUNCTIONS
IN TERMS OF RIGHT TRIANGLES.
IF YOU NEED TO BUILD A RAMP THAT HAS AN INCLINE OF 6 DEGREES
AND MUST REACH A HEIGHT OF 2 FEET,
HOW LONG WOULD THE RAMP HAVE TO BE?
WELL, WE CAN SOLVE THIS TYPE OF PROBLEM USING TRIGONOMETRY.
THERE ARE TRIGONOMETRIC FUNCTIONS
AND THEY ARE SIGN, COSIGN, TANGENT, COSECANT, SECANT,
AND COTANGENT.
CUSTOMARILY, THEY ARE ABBREVIATED
USING THIS NOTATION IN BOLD.
NOW, THESE TRIG FUNCTIONS CAN BE DEFINED
IN TERMS OF PARTICULAR RATIOS OF SIDES OF RIGHT TRIANGLES.
HERE ARE THE SIGN, COSIGN, AND TANGENT FUNCTIONS
IN TERMS OF RATIOS OF SIDES OF RIGHT TRIANGLES.
IF WE START AT ANGLE "A" IT WOULD BE THE RATIO
OF THE OPPOSITE SIDE OVER THE HYPOTENUSE.
SO LETS BACK UP HERE FOR A SECOND AND IDENTIFY.
REMEMBER THE HYPOTENUSE IS ALWAYS OPPOSITE THE RIGHT ANGLE,
SO IF YOU WERE TO BISECT THE RIGHT ANGLE,
THE SIDE THAT YOU RUN INTO WOULD ALWAYS BE THE HYPOTENUSE.
NOW, THE OPPOSITE AND ADJACENT SIDES WILL CHANGE
BASED UPON WHICH ANGLE YOU'RE REFERRING TO.
SO GOING BACK TO ANGLE "A," IF WE BISECT ANGLE "A"
THE SIDE THAT WE RUN INTO WOULD BE THE OPPOSITE SIDE.
SO IN TERMS OF ANGLE "A" THIS IS THE OPPOSITE SIDE.
SO THE SIGN OF "A" IS THE RATIO OF THE OPPOSITE
OVER THE HYPOTENUSE SIDE.
SO IN THIS CASE IT WOULD BE A/C.
LET'S STICK WITH ANGLE "A" FOR A MOMENT.
THE COSIGN OF ANGLE "A" WOULD BE THE RATIO OF THE ADJACENT SIDE
OVER THE HYPOTENUSE.
WELL, IF THIS IS THE OPPOSITE SIDE FROM ANGLE "A,"
THIS WOULD BE THE ADJACENT SIDE
BECAUSE THIS IS ALWAYS THE HYPOTENUSE.
SO THE COSIGN OF "A" WOULD BE THE RATIO OF THE ADJACENT SIDE
OVER THE HYPOTENUSE, OR B/C.
AND TANGENT OF ANGLE "A" WOULD BE THE RATIO
OF THE OPPOSITE SIDE OVER THE ADJACENT SIDE.
SO HERE WE'D HAVE A/B.
OKAY, NOW IF WE LOOK AT THE SECOND ROW,
NOTICE NOW REFER TO ANGLE B,
SO THAT'S GOING TO CHANGE THE ORIENTATION OF THE OPPOSITE
AND ADJACENT SIDES.
SO IF WE'RE REFERRING TO ANGLE B, IF WE WERE TO BISECT THIS,
NOW ANGLE B WOULD BE THE OPPOSITE SIDE,
AND ANGLE "A" WOULD BE THE ADJACENT SIDE.
SO NOTICE HOW REGARDLESS OF WHAT ANGLE WE REFER TO
THE RATIO IS ALWAYS THE SAME.
SIGN IS THE RATIO OF THE OPPOSITE OVER THE HYPOTENUSE.
COSIGN IS ADJACENT OVER HYPOTENUSE.
AND TANGENT IS ALWAYS OPPOSITE OVER ADJACENT.
SO LOOKING AT ANGLE B NOW,
WE HAVE OPPOSITE OVER HYPOTENUSE WOULD BE B/C FOR THE SIGN B.
COSIGN B WOULD BE ADJACENT OVER HYPOTENUSE OR A/C.
AND THEN TANGENT OF B WOULD BE OPPOSITE OVER ADJACENT OR B/A.
ONE WAY TO REMEMBER ALL OF THESE
WOULD BE TO JUST REMEMBER THIS ACRONYM S.O.H.C.A.H.T.O.A.,
WHICH STANDS FOR THE SIGN IS OPPOSITE OVER HYPOTENUSE,
COSIGN IS ADJACENT OVER HYPOTENUSE,
AND TANGENT IS OPPOSITE OVER ADJACENT.
NOW, BEFORE WE LEARN THAT ALL RIGHT TRIANGLES
WITH THE SAME ACUTE ANGLES ARE SIMILAR,
AND SO IF WE HAVE THAT CASE,
THE TRIGONOMETRIC FUNCTIONS WILL PRODUCE THE SAME RATIOS
BECAUSE WE KNOW THAT THESE TRIANGLES ARE PROPORTIONAL.
SO, AGAIN, IF YOU TAKE A LOOK AT THE LARGE RIGHT TRIANGLE
AND THE SMALL GREEN TRIANGLE, THEY SHARE A COMMON ANGLE HERE,
AND THEY ALSO HAVE TWO RIGHT ANGLES.
SO BASED UPON ANGLE, ANGLE,
THESE ARE SIMILAR RIGHT TRIANGLES.
SO LET'S GO AHEAD AND LABEL THIS FULL LENGTH.
THAT'S 5 AND THAT'S 5, THIS WOULD BE 10.
AND IF THIS IS 4, AND THIS IS 4, THIS WOULD BE 8.
SO FOR THE SIGN "A" YOU WOULD HAVE THE OPPOSITE LENGTH 6
OVER THE HYPOTENUSE LIKE 10, 6/10.
NOW, IF WE LOOK AT THE SMALL TRIANGLE,
OPPOSITE OVER HYPOTENUSE WOULD BE 3/5.
AND WE CAN SEE HERE THAT 6/10 IS EQUIVALENT TO 3/5
PRODUCING THE SAME RATIO.
THE SAME FOR THE COSIGN OF ANGLE "A".
AND THE LARGE TRIANGLE YOU WOULD HAVE 8/10.
AND IN THE SMALL TRIANGLE YOU WOULD HAVE 4/5,
WHICH AGAIN ARE EQUIVALENT RATIOS.
AND FOR THE TANGENT OF "A"
WE WOULD HAVE THE OPPOSITE OVER THE ADJACENT
OR 6 TO 8 FOR THE LARGE RIGHT TRIANGLE,
AND THE SMALL WOULD BE 3/4, AGAIN EQUIVALENT.
NOW, LET'S GO AHEAD
AND DEFINE THE THREE OTHER TRIGONOMETRIC FUNCTIONS,
AND THOSE ARE COSECANT, SECANT, AND COTANGENT.
AND WHAT I WANT YOU TO NOTICE HERE,
IS THESE ARE RECIPROCALS OF THE SIGN, COSIGN,
AND TANGENT FUNCTIONS.
SO COSECANT "A" WOULD BE THE RATIO OF THE HYPOTENUSE
TO THE OPPOSITE SIDE.
REMEMBER THE SIGN WAS OPPOSITE OVER HYPOTENUSE.
AND THE SECANT OF "A" WOULD BE HYPOTENUSE OVER ADJACENT,
WHICH IS RECIPROCAL OF ADJACENT OVER HYPOTENUSE,
WHICH IS COSIGN.
AND COTANGENT IS ADJACENT OVER OPPOSITE,
WHILE TANGENT IS OPPOSITE OVER ADJACENT.
SO, AGAIN, ALL THESE ARE REFER TO ANGLE "A" HERE.
SO THIS WOULD BE OUR HYPOTENUSE.
"A" WOULD BE THE OPPOSITE AND B WOULD BE THE ADJACENT.
SO WE CAN SEE HERE FOR COSECANT "A"
WE HAVE THE HYPOTENUSE OVER THE OPPOSITE OR C/A.
SECANT WOULD BE HYPOTENUSE OVER ADJACENT, C/B.
AND COTANGENT WOULD BE ADJACENT OVER OPPOSITE OR B/A.
LET'S DETERMINE THE SIGN, COSIGN,
AND TANGENT OF ANGLE "A".
SO HERE'S OUR ANGLE "A".
LET'S GO AHEAD AND IDENTIFY THE THREE SIDES.
AGAIN, THIS IS THE HYPOTENUSE.
IF THIS IS ANGLE "A," WE BISECT IT,
THIS WOULD BE THE OPPOSITE SIDE,
AND THIS LEAVES THIS SIDE AS THE ADJACENT SIDE.
SO, AGAIN, IF WE WANT TO,
WE CAN USE THE ACRONYM S.O.H.C.A.H.T.O.A.
IF THAT'S HELPFUL.
NOW, BEFORE WE FIND THESE THREE VALUES,
I NOTICE THAT THE LENGTH OF THE HYPOTENUSE IS MISSING.
SO WE'RE GOING TO HAVE TO APPLY THE PYTHAGOREAN THEOREM
TO FIND THIS LENGTH.
REMEMBER C SQUARED = A SQUARED + B SQUARED
WHERE C IS THE HYPOTENUSE.
SO WE WOULD HAVE C SQUARED = 5 SQUARED + 12 SQUARED.
SO WOULD BE 25 + 144, C SQUARED = 169, SO C = 13.
YOU MAY'VE RECOGNIZED THIS AS A 5-12-13 RIGHT TRIANGLE.
SO THE SIGN OF ANGLE "A" WOULD BE THE OPPOSITE
OVER THE HYPOTENUSE, SO 12/13.
THE COSIGN OF ANGLE "A," AGAIN IS ADJACENT OVER HYPOTENUSE,
SO THAT WOULD 5/13.
AND TANGENT OF "A" WOULD BE OPPOSITE OVER ADJACENT OR 12/5.
AND THAT'S IT, WE FOUND THOSE TRIGONOMETRIC VALUES.
LET'S TRY ONE MORE.
AGAIN, I CAN SEE RIGHT AWAY
WE'RE MISSING THE LENGTH OF THIS THIRD SIDE.
GOING BACK TO OUR PYTHAGOREAN THEOREM, THIS IS OUR HYPOTENUSE,
SO WE'D HAVE 8 SQUARED = 4 SQUARED + 8 SQUARED.
SO IF WE SUBTRACT 16 ON BOTH SIDES WE'D HAVE 48.
AND SQUARE ROOT BOTH SIDES.
WE ONLY WANT THE PRINCIPAL SQUARE ROOT,
SO WE'D HAVE X = THE SQUARE ROOT OF 48.
BUT REMEMBER 48 IS 16 x 3,
AND THE SQUARE ROOT OF 16 WOULD BE 4,
SO THAT'S EQUAL TO 4 SQUARE ROOT 3.
OKAY. LET'S GO AHEAD AND SET THIS UP.
NOW, THEY ARE ASKING FOR COSECANT, SECANT,
AND COTANGENT OF ANGLE B,
SO HERE'S THE ANGLE WE'RE REFERRING TO.
SO THE 4 WOULD BE THE OPPOSITE
AND 4 SQUARE ROOT 3 WOULD BE THE ADJACENT SIDE.
NOW, WE CAN STILL USE THE ACRONYM S.O.H.C.A.H.T.O.A.
TO HELP US AS LONG AS YOU KNOW
WHICH FUNCTION THESE ARE THE RECIPROCALS OF.
COSECANT IS THE RECIPROCAL OF THE SIGN FUNCTION,
SO SIGN IS OPPOSITE OF OUR HYPOTENUSE.
COSECANT WOULD BE HYPOTENUSE OVER OPPOSITE.
WELL, STARTING AT ANGLE B, THE HYPOTENUSE IS 8
AND THE OPPOSITE IS 4,
SO WE WOULD HAVE 8/4, WHICH EQUALS 2.
SECANT IS THE RECIPROCAL OF COSIGN,
SO IF COSIGN IS ADJACENT OVER HYPOTENUSE,
WE WANT THE HYPOTENUSE OVER THE ADJACENT,
WHICH WOULD BE 8/4 SQUARE ROOT 3.
NOW, THIS SIMPLIFIES TO 2 OVER SQUARE ROOT 3.
YOU MAY BE REQUIRED TO RATIONALIZE THIS.
THIS RATIONALIZE TO 2 SQUARE ROOT 3/3.
AND LASTLY, COTANGENT IS THE RECIPROCAL OF TANGENT,
SO WE WOULD WANT THE ADJACENT OVER THE OPPOSITE,
WHICH WOULD GIVE US 4 SQUARE ROOT 3/4,
WHICH JUST SIMPLIFIES TO SQUARE ROOT 3.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
HAVE A GOOD DAY.