Trigonometric identity By: EDU-MASTER Classes MAGIC HEXAGON STEP 1: DRAW A HEXAGON STEP 2: Join all the opposite vertices 1.Now here we get three diagonals 2. Lets give 1 to the center STEP 3: How to give the six function name to each of vertices in particular way 1. Give function name from this point in a clock- wise direction. 2. End at this point. STEP 4: Now lets write the six function names at vertices. sin We just remember one formula which we have learn. π‘ππ=π ππ/πππ 2. Lets write tan at this point. 3. Now write sin and cos in clock wise direction. 4. Now on remaining vertices we are going to write cot opposite to tan. tan cos cot sin 5. If we draw the vertical line from center of hexagon. 6. Then we are having three vertices on left and three on right. 7. Now remember All C’s should be on the right side. 8. Therefore write cosec on this point and sec on remaining. tan sec cos cot cosec STEP 5: Lets calculate formulas. sin cos 1. Start from tan go clock-wise we will get π πππ πππ π tan cot tan π = 2. Lets repeat the procedure start from sin use next two functions from sin we will get Sin π = sec cosec cosπ cotπ sin tan Now do it for all functions And we get this cos cotπ cosπ = cosecπ cot cosecπ cotπ = π eππ secπ cosecπ = tanπ sec cosec tanπ secπ = sinπ sin What if we go anti clock wise? Don’t worry this will also work. Lets see cos tan cot sec cosec 3. secπ 1. tanπ = cosecπ 2. π ecπ = cosecπ cotπ cosecπ = cotπ cosπ cosπ sinπ 4. cotπ = 5. cosπ = tanπ 6. sinπ = sinπ tanπ π eππ Clockwise anticlockwise secπ cosecπ tan π = π πππ πππ π tanπ = Sin π = cosπ cotπ cosecπ π ecπ = cotπ cotπ cosπ = cosecπ cosecπ = cotπ cos π cosπ sinπ cosecπ cotπ = π eππ cotπ = secπ cosecπ = tanπ sinπ cosπ = tanπ tanπ secπ = sinπ sinπ = tanπ π eππ Lets see why 1 is placed in centre? sin cos 1. 2. tan cot We are going to multiply opposite vertices and we will get 1 As follows: sin π . cosecπ = 1 cosπ. π πππ = 1 tanπ. cotπ = 1 sec cosec sin cos 2. Lets take any three continues function For example: tan, sin, cos then if we multiply we will get tanπ π₯ cosπ = sinπ tan sec cot cosec Hence we will get sin π × cot π = cos π cos ππ₯ cosec π = cot π cot ππ₯ sec π = cosec π cos eπππ₯ tan π = sec π π eπ × sin π = tan π sin 3. Now cos and sec are reciprocal of each other that’s why we get cos 1 cos π = sec π tan cot 1 sin π = cosec π 1 tan π = cot π sec cosec Like this we will get total six formulas 4. What is complimentary angle? sin cos tan cot sec cosec Yes,Two Angles are Complementary when they add up to 90 degrees (a Right Angle). And trigonometric function has special relationship with complementary angle. Lets see this tells us sin π = cos 900 − π hence sinπ is complimentry to cosπ. Similarly sin cos tanπ=cot(900 − π) πππ π πππ = (900 − π) tan cot sec cosec If we reverses the arow we will get sin cos cot π = tan 900 − π tan sec cot cosec cosec π = sec 900 − π cos π = sin 900 − π 5. Now look at the hexagon we get six triangles inside it. sin tan sec cos cot cosec Focus on three of triangles sin cos Now we will go clockwise sin, cos and 1 This gives us idea, π ππ2 π+ πππ 2 π = 1 tan cot Now next triangle 1 + πππ‘ 2 π = πππ ππ 2 π And 3rd triangle tan2 π + 1 = sec 2 π sec cosec We can also go anti-clock wise the only change is we have to put minus sign before second term. sin cos 1-πππ 2 π = π ππ2 π πππ ππ 2 π − πππ‘ 2 π = 1 tan sec cot cosec π ππ 2 π -1= π‘ππ2 π
