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SOLUTIONS TO TRIANGLES

SOLUTIONS TO TRIANGLES. PURPOSE. as we studied in the last chapter “trigonometry” literally means measurement of triangles. So in the present lecture we will develop formulas using which we can find the unknown sides and angles of a triangle. SOME STANDARD DENOTIONS.

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SOLUTIONS TO TRIANGLES

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  1. SOLUTIONS TO TRIANGLES

  2. PURPOSE as we studied in the last chapter “trigonometry” literally means measurement of triangles So in the present lecture we will develop formulas using which we can find the unknown sides and angles of a triangle

  3. SOME STANDARD DENOTIONS In all the following formulas and concepts in the present lecture, the understated denotions will be followed

  4. BEFORE WE START, AN IMPORTANT NOTE SYMMETRY The concept of symmetry will be very useful in our present discussion by using which we can generalize a formula derived for a certain side or angle of a triangle to all the three sides or angles of the same triangle

  5. THE SINE FORMULA

  6. SIMILARLY WE CAN DO THE SAME FOR B AND C ALSO finally we will obtain what is known as the sine formula

  7. THE COSINE FORMULA It is derived using the pythagoras theorem and has 3 symmetrical forms

  8. projection FORMULAs PROOF :

  9. SIMILARLY WE CAN PROVE :

  10. HALF ANGLE FORMULAES Here ‘s’ is the semi-perimeter we start with the cosine formula PROOF :

  11. SIMILARLY WE CAN PROVE ALL THE OTHER FORMULAS FOLLOW FROM SYMMETRY

  12. NAPIER’S ANALOGIES WE CAN USE THE COMPOUND ANGLE FORMULA FOR tan(a+b) AND THE HALF ANGLE FORMULA

  13. AREA OF TRIANGLE (Δ) Using this we can derive Which again by symmetry is : and

  14. HERO’S FORMULA WE ALREADY KNOW WE EXPAND sinA USING MULTIPLE ANGLE FORMULA Now we use the half angle formulas which we previously derived

  15. So we get : and finally we have the hero’s formula as : WHERE ‘R’ IS THE CIRCUMRADIUS AND ‘r’ IS THE INRADIUS

  16. A SIMPLE EXAMPLE PROVE THAT : By looking at the RHS we can say that using the cosine formula will be beneficial as it will give us squares of sides Using sine formula and cosine formula

  17. = RHS, thus proved

  18. ANOTHER ONE !!

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