Transcript for: Cofunction Identities in Trigonometry
WELCOME TO A LESSON ON COFUNCTION IDENTITIES. BEFORE WE DISCUSS THE IDENTITIES, THOUGH, I WOULD LIKE TO DO A QUICK REVIEW SO THAT WE CAN BETTER UNDERSTAND WHY THE IDENTITIES MAKE SENSE. SO FIRST, THE SUM OF THE INTERIOR ANGLES OF ANY TRIANGLE IS 180 DEGREES OR PI RADIANS. SO WE CAN SAY THE SUM OF THE MEASURE OF ANGLES "A," B, AND C IS 180 DEGREES OR PI RADIANS, AND THEN THE TWO ACUTE ANGLES OF A RIGHT TRIANGLE HAVE A SUM OF 90 DEGREES OR PI/2 RADIANS, WHICH MEANS THE TWO ACUTE ANGLES OF A RIGHT TRIANGLE ARE COMPLIMENTARY. SO AGAIN, IF WE KNOW THE SUM IS EQUAL TO 180 DEGREES, BUT THE MEASURE OF ANGLE C IS EQUAL TO 90 DEGREES, THAT MEANS THE MEASURE OF ANGLE "A" PLUS THE MEASURE OF ANGLE B MUST EQUAL 90 DEGREES OR PI/2 RADIANS, AND THEREFORE, ANGLE "A" AND ANGLE B ARE COMPLIMENTARY. WE ALSO NEED TO BE FAMILIAR WITH THE RIGHT TRIANGLE DEFINITIONS OF THE SIX TRIG FUNCTIONS GIVEN HERE. SO IF YOU NEED TO, YOU MAY WANT TO PAUSE HERE. LET'S GO AHEAD AND TALK ABOUT COFUNCTION IDENTITIES. A FUNCTION F IS A COFUNCTION OF A FUNCTION G IF F OF "A" EQUALS G OF B WHEN A AND B ARE COMPLIMENTARY ANGLES. SO FUNCTION F AND G ARE DIFFERENT FUNCTIONS, BUT THEY'RE EQUAL TO EACH OTHER AS LONG AS ANGLE "A" AND ANGLE B ARE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES. SO MOST OF THE TIME YOU'LL SEE COFUNCTION IDENTITIES GIVEN IN THE FORM AS WE SEE HERE IN BLUE. FIRST, WE HAVE THE SINE OF ANGLE "A" = THE COSINE OF THE QUANTITY 90 DEGREES - "A," AND ALSO, COSINE "A" = SINE OF THE QUANTITY 90 DEGREES - "A." WHAT THIS IDENTITY IS TRYING TO TELL US IS THAT THESE TWO TRIG FUNCTIONS ARE EQUAL AS LONG AS THE ANGLES ARE COMPLIMENTARY. ANGLE "A" AND THE ANGLE THAT MEASURES 90 DEGREES - "A" ARE COMPLIMENTARY ANGLES. AND THAT'S WHY I THINK IT'S OFTEN HELPFUL TO EXPRESS THE COFUNCTION IDENTITY AS SINE "A" = COSINE B IF "A" + B = 90 DEGREES OR PI/2 RADIANS. THIS EQUATION HERE TAKES THE PLACE OF THESE TWO EQUATIONS HERE, AND THEN AGAIN, INSTEAD OF USING THESE TWO EQUATIONS, WE CAN SAY SECANT "A" = COSECANT B IF "A" + B = 90 DEGREES OR THE TWO ANGLES "A" AND B ARE COMPLIMENTARY. AND THE SAME THING FOR TANGENT AND COTANGENT. TANGENT "A" = COTANGENT B IF "A" + B = 90 DEGREES. NOTICE HOW THESE ARE ALL EXPRESSED IN DEGREES, BUT SINCE PI/2 RADIANS IS EQUAL TO 90 DEGREES, WE CAN ALSO EXPRESS THESE IDENTITIES IN THIS FORM HERE. LET'S GO BACK AND TAKE A LOOK AT WHY THESE IDENTITIES MAKE SENSE. IF WE TAKE A LOOK AT THIS GREEN RIGHT TRIANGLE HERE, LET'S GO AHEAD AND JUST ASSUME THE MEASURE OF ANGLE "A" IS 50 DEGREES. BUT WE SHOULD RECOGNIZE THAT ANGLE "A" AND ANGLE B ARE COMPLIMENTARY, MEANING THEY'D HAVE A SUM OF 90 DEGREES. THEREFORE, THE MEASURE OF ANGLE B MUST BE 40 DEGREES. NOW, LET'S COMPARE THE SINE OF ANGLE "A" OR THE SINE OF 50 DEGREES TO THE COSINE OF 40 DEGREES. AGAIN, NOTICE HOW THESE TWO ANGLES ARE COMPLIMENTARY. WELL, THE SINE OF ANGLE "A" WOULD BE THE RATIO OF THE LENGTHS OF THE OPPOSITE SIDE TO THE HYPOTENUSE OR IN THIS CASE, A/C. WELL, NOW IF WE SWITCH TO ANGLE B AND FIND THE COSINE OF ANGLE B, WHICH AGAIN, IS 40 DEGREES, IT'S GOING TO BE THE RATIO OF THE LENGTH OF THE ADJACENT SIDE TO THE HYPOTENUSE, WHICH AGAIN, NOTICE IS A/C. NOTICE THE ADJACENT SIDE TO ANGLE B IS THE SAME AS THE OPPOSITE SIDE OF ANGLE "A," AND, THEREFORE, THESE TWO TRIG FUNCTION VALUES ARE EQUAL TO EACH OTHER, AGAIN, AS LONG AS THE TWO ANGLES ARE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES. WE CAN ALSO VERIFY THE OTHER IDENTITIES IN A SIMILAR WAY. LET'S GO AHEAD AND TAKE A LOOK AT OUR EXAMPLES. WE WANT TO WRITE EACH FUNCTION IN TERMS OF ITS COFUNCTION. SO WE HAVE THE SINE OF 18 DEGREES. WELL, THE COFUNCTION IDENTITY FOR SINE INVOLVES COSINE. SO THE SINE OF 18 DEGREES IS EQUAL TO THE COSINE OF-- LET'S GO AHEAD AND CALL IT ANGLE B. AND, AGAIN, THE MAIN THING TO REMEMBER HERE IS THAT THESE TWO ANGLES MUST BE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES. SO ANGLE B + 18 DEGREES MUST EQUAL 90 DEGREES. SO IF WE SUBTRACT 18 DEGREES ON BOTH SIDES, WE CAN SEE THAT ANGLE B MUST BE 72 DEGREES. THEREFORE, THE SINE OF 18 DEGREES IS EQUAL TO THE COSINE OF 72 DEGREES. AND NOTICE WE COULD'VE USED THIS IDENTITY HERE AND JUST REPLACED "A" WITH 18 DEGREES, GIVING US 72 DEGREES AS WELL. BUT I PREFER TO APPROACH IT THIS WAY TO EMPHASIZE THAT THESE TWO ANGLES MUST BE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES. FOR OUR SECOND EXAMPLE WE HAVE TANGENT 65 DEGREES. THAT'S GOING TO BE EQUAL TO COTANGENT OF, AGAIN, LET'S CALL IT ANGLE B WHERE THESE TWO ANGLES ARE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES, WHICH MEANS B + 65 DEGREES MUST EQUAL 90 DEGREES. SUBTRACTING 65 DEGREES ON BOTH SIDES, WE CAN SEE THAT B IS GOING TO BE EQUAL TO 25 DEGREES. THEREFORE, TANGENT 65 DEGREES MUST EQUAL COTANGENT OF 25 DEGREES. NEXT WE HAVE COSECANT 84 DEGREES. THE COFUNCTION IDENTITY FOR A COSECANT INVOLVES SECANT OF ANGLE B WHERE, AGAIN, THESE TWO ANGLES ARE COMPLIMENTARY OR HAVE A SUM OF 90 DEGREES. SO WE CAN PROBABLY TELL BY INSPECTION THAT ANGLE B IS GOING TO BE 6 DEGREES. SO WE HAVE COSECANT 84 DEGREES MUST EQUAL SECANT 6 DEGREES. BEFORE WE TAKE A LOOK AT SOME INVOLVING RADIANS, LET'S VERIFY THESE ON OUR CALCULATOR. LET'S FIRST MAKE SURE THAT WE'RE IN DEGREE MODE. SO WE'LL PRESS THE MODE KEY, GO DOWN TO THE THIRD ROW AND HIGHLIGHT DEGREE, PRESS ENTER, GO BACK TO THE HOME SCREEN, 2nd, QUIT, WE'LL TYPE IN SINE 18 DEGREES, ENTER, WE'LL COMPARE THAT TO COSINE 72 DEGREES, ENTER, AND NOTICE HOW THEY ARE EQUAL TO EACH OTHER. AND THEN WE HAVE TANGENT 65 DEGREES AND COTANGENT 25 DEGREES. NOW, REMEMBER, THERE'S NO COTANGENT KEY BUT COTANGENT THETA IS EQUAL TO 1/TANGENT THETA. SO I'M GOING TO TYPE IN 1 DIVIDED BY, AND THEN IN PARENTHESES, TANGENT 25 DEGREES. AGAIN, THIS IS EQUAL TO COTANGENT 25 DEGREES, AND WE CAN SEE THEY ARE EQUAL. SO I'LL GO AHEAD AND STOP HERE, BUT WE CAN ALWAYS VERIFY THESE ON A CALCULATOR. LET'S TAKE A LOOK AT THREE MORE EXAMPLES THAT INVOLVE RADIANS INSTEAD OF DEGREES. AGAIN, THE COFUNCTION IDENTITY INVOLVING COSINE ALSO INVOLVES SINE. IT'S GOING TO BE EQUAL TO SINE OF ANGLE B WHERE THESE TWO ANGLES ARE COMPLIMENTARY OR IN THIS CASE, HAVE A SUM OF PI/2 RADIANS. SO B + PI/4 MUST EQUAL PI/2. SO HERE WE'LL ACTUALLY SHOW SOME WORK. WE'LL GO AHEAD AND SUBTRACT PI/4 RADIANS ON BOTH SIDES OF THE EQUATION. THIS WOULD BE ZERO. SO WE HAVE B EQUALS THIS DIFFERENCE HERE, BUT WE DO HAVE TO HAVE A COMMON DENOMINATOR. SO WE'RE GOING TO HAVE TO MULTIPLY THIS FRACTION BY 2/2. SO NOW WE HAVE 2PI/4 AND HERE WE HAVE MINUS 1PI/4. WELL, 2PI/4 - 1PI/4 = 1PI/4 OR JUST PI/4. SO WE HAVE COSINE PI/4 = SINE PI/4. LET'S JUST TAKE A MOMENT AND TAKE A LOOK AT THIS ON THE UNIT CIRCLE. HERE IS PI/4 RADIANS OR 45 DEGREES. NOTICE ON THE UNIT CIRCLE THE X AND Y COORDINATES ARE THE SAME, VERIFYING THAT WHEN THETA IS PI/4, COSINE AND SINE ARE EQUAL TO EACH OTHER. NEXT, WE HAVE COTANGENT PI/3. THAT'S GOING TO BE EQUAL TO TANGENT OF ANGLE B WHERE, AGAIN, THESE TWO ANGLES ARE COMPLIMENTARY OR HAVE A SUM OF PI/2 RADIANS. SO B + PI/3 MUST EQUAL PI/2. SUBTRACT PI/3 ON BOTH SIDES. SO WE HAVE ANGLE B MUST EQUAL THIS DIFFERENCE HERE. OUR COMMON DENOMINATOR HERE IS GOING TO BE 6. SO WE'LL MULTIPLY THIS BY 3/3 AND THIS BY 2/2. SO NOW WE HAVE 3PI/6 - 2PI/6, WHICH IS GOING TO GIVE US 1PI/6 OR JUST PI/6. SO WE HAVE COTANGENT PI/3 = TANGENT OF PI/6. AND THEN FOR OUR LAST EXAMPLE WE HAVE SECANT PI/6. THIS COFUNCTION IDENTITY IS GOING TO INVOLVE COSECANT OF THE ANGLE THAT'S COMPLIMENTARY WITH PI/6, BUT WE CAN TELL FROM EXAMPLE TWO, PI/6 AND PI/3 ARE COMPLIMENTARY OR HAVE A SUM OF PI/2 RADIANS. THEREFORE, THIS IS GOING TO BE COSECANT PI/3 RADIANS. AND, AGAIN, WE CAN VERIFY THESE ON THE GRAPHING CALCULATOR OR BECAUSE ALL OF THESE ARE NICE REFERENCE ANGLES ON THE UNIT CIRCLE, WE COULD VERIFY THESE WITH THE UNIT CIRCLE AS WELL. SO YOU MAY WANT TO PAUSE THE VIDEO HERE AND VERIFY THAT THESE IDENTITIES HERE DO HOLD TRUE. OKAY, I HOPE YOU HAVE FOUND THIS HELPFUL. NEXT, WE'LL TAKE A LOOK AT SOLVING EQUATIONS INVOLVING COFUNCTION IDENTITIES.