- WE WANT TO FIND THE EXACT TRIG FUNCTION VALUES
USING THE RIGHT TRIANGLE.
BECAUSE WE WANT TO FIND ALL SIX TRIG FUNCTION VALUES,
WE NEED THE LENGTH OF ALL THREE SIDES OF A TRIANGLE.
NOTICE HOW WE'RE MISSING A LENGTH OF THE HYPOTENUSE
OR SIDE C.
SO THE FIRST STEP IS TO DETERMINE THE LENGTH OF SIDE C
USING THE PYTHAGOREAN THEOREM.
SO USING C SQUARED = A SQUARED + B SQUARED,
WHERE SIDE "A" HAS THE LENGTH OF 6
AND SIDE B HAS THE LENGTH OF 3.
WE WOULD HAVE C SQUARED = 6 SQUARED + 3 SQUARED,
SO WE'D HAVE C SQUARED = 36 + 9.
SO C SQUARED = 45.
WE KNOW C IS A LENGTH, SO IT HAS TO BE POSITIVE.
SO WE'LL ONLY CONSIDER THE POSITIVE SOLUTION HERE.
C IS EQUAL TO THE SQUARE ROOT OF 45,
BUT 45 IS EQUAL TO 9 x 5.
SINCE THE SQUARE ROOT OF 9 IS EQUAL TO 3
WE HAVE 3 SQUARE ROOT 5 FOR THE LENGTH OF THE HYPOTENUSE.
NOW WE CAN FIND ALL SIX TRIG FUNCTION VALUES,
BUT FOR A QUICK REVIEW,
THE SIGN OF AN ANGLE IS EQUAL TO THE RATIO OF THE LENGTH
OF THE OPPOSITE SIDE OF THE HYPOTENUSE.
AND SINCE SIGN AND COSECANT ARE RECIPROCALS,
ONCE WE FIND THE SIGN FUNCTION VALUE,
WE CAN TAKE THE RECIPROCAL
TO FIND THE COSECANT FUNCTION VALUE,
WHICH IS EQUAL TO THE RATIO OF THE HYPOTENUSE
TO THE OPPOSITE SIDE.
THE COSIGN OF AN ANGLE IS EQUAL TO THE RATIO
OF THE ADJACENT SIDE OF THE HYPOTENUSE.
AND, AGAIN, SINCE COSIGN AND SECANT ARE RECIPROCALS,
SECANT OF AN ANGLE IS EQUAL TO THE RATIO OF THE HYPOTENUSE
TO THE ADJACENT SIDE.
AND THEN FINALLY, THE TANGENT OF AN ANGLE
IS EQUAL TO THE RATIO OF THE OPPOSITE SIDE
TO THE ADJACENT SIDE.
AND THEN SINCE COTANGENT AND TANGENT ARE RECIPROCALS,
COTANGENT OF AN ANGLE IS EQUAL TO THE RATIO
OF THE ADJACENT SIDE TO THE OPPOSITE SIDE.
THE ONLY THING ABOUT THIS IS,
IF WE CAN FIND THE SIGN FUNCTION VALUE,
COSIGN FUNCTION VALUE, AND TANGENT FUNCTION VALUES,
THEN WE CAN JUST FIND THE RECIPROCALS
TO FIND THE OTHER THREE TRIG FUNCTION VALUES.
BECAUSE ALL THESE REFERENCE ANGLE B,
SIDE B WOULD BE THE OPPOSITE SIDE.
THE HYPOTENUSE IS SIDE C.
AND SIDE "A" WILL BE THE ADJACENT SIDE.
NOW WE CAN FIND ALL SIX TRIG FUNCTION VALUES.
THE SIGN OF ANGLE B IS THE RATIO
OF THE OPPOSITE SIDE OF THE HYPOTENUSE,
SO WE WOULD HAVE 3/3 SQUARE ROOT 5
OR 3 DIVIDED BY 3 SQUARE ROOT 5.
NOTICE HOW THIS SIMPLIFIES.
3/3 SIMPLIFIES TO 1,
SO THE EXACT SIGN FUNCTION VALUE IS 1
DIVIDED BY SQUARE ROOT 5.
BUT SOMETIMES WE ARE ASKED TO RATIONALIZE THESE VALUES,
SO LET'S GO AHEAD AND DO THAT AS WELL.
WE'D MULTIPLY BY SQUARE ROOT 5 OVER SQUARE ROOT 5,
SO THIS WOULD BE EQUAL TO SQUARE ROOT 5 OVER--
WELL, SQUARE ROOT 5 AND SQUARE ROOT 5 IS JUST 5.
SO, AGAIN, THE VALUE CAN BE EXPRESSED TWO WAYS,
1 DIVIDED BY SQUARE ROOT 5,
OR IF WE'RE REQUIRED TO RATIONALIZE THE VALUE,
IT WOULD BE SQUARE ROOT 5/5.
AGAIN, BECAUSE THE COSECANT OF ANGLE B
IS THE RECIPROCAL OF THE SIGN OF ANGLE B,
WE CAN USE RECIPROCAL OF 1 OVER SQUARE ROOT 5
TO FIND THIS TRIG FUNCTION VALUE
THAT WOULD JUST BE SQUARE ROOT 5/1 OR JUST SQUARE ROOT 5.
IF WE WANTED TO USE THE RIGHT TRIANGLE,
IT WOULD BE THE RATIO OF THE HYPOTENUSE
TO THE OPPOSITE SIDE,
WHICH WOULD BE 3 SQUARE ROOT 5 DIVIDED BY 3,
WHICH IS STILL SQUARE ROOT 5.
NEXT, COSIGN OF ANGLE B WILL BE THE RATIO
OF THE ADJACENT SIDE TO THE HYPOTENUSE,
WHICH WOULD BE 6 DIVIDED BY 3 SQUARE ROOT 5,
WHICH DOES SIMPLIFY.
WE WOULD HAVE 2 DIVIDED BY SQUARE ROOT 5,
WHICH IS THE EXACT VALUE OF COSIGN B.
BUT, AGAIN, IF WE'RE REQUIRED TO RATIONALIZE,
LET'S GO AHEAD AND DO THAT.
WE WOULD HAVE 2 SQUARE ROOT 5/5.
THE SECANT OF ANGLE B WILL BE THE RECIPROCAL
OF THE COSIGN OF ANGLE B.
WELL, THE RECIPROCAL OF 2 OVER SQUARE ROOT 5
WOULD BE SQUARE ROOT 5/2.
OR USING THE RIGHT TRIANGLE,
IT WOULD BE THE RATIO OF THE HYPOTENUSE
TO THE ADJACENT SIDE
OR 3 SQUARE ROOT 5 DIVIDED BY 6,
WHICH WOULD STILL BE SQUARE ROOT 5/2.
THE TANGENT OF ANGLE B WILL BE THE RATIO OF THE OPPOSITE SIDE
TO THE ADJACENT SIDE,
WHICH WOULD BE 3 DIVIDED BY 6 OR 3/6,
WHICH SIMPLIFIES TO 1/2.
AND THE COTANGENT OF B WILL BE THE RECIPROCAL OF TANGENT B,
SO THIS TRIG FUNCTION VALUE MUST BE 2/1 OR JUST 2.
OR USING THE RIGHT TRIANGLE,
IT WILL BE THE RATIO OF THE ADJACENT SIDE
TO THE OPPOSITE SIDE
WHERE 3 DIVIDED BY 6, WHICH DOES EQUAL 2.
OKAY. THAT'S GOING TO DO IT FOR THIS EXAMPLE.
I HOPE YOU FOUND THIS HELPFUL.