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t3 analytic trigonometry and trig formulas

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math260

The document defines several trigonometric reciprocal functions including cosecant, secant, and cotangent. It then presents trigonometric identities relating these functions through division, reciprocal, and square-sum relations. These fundamental identities are considered important relationships that hold for all angles and are used throughout analytic trigonometry.

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Analytic Trigonometry
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown.
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown. (cos(),sin())y=sin()(1,0)x=cos()
Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown. By Pythagorean Theorem,y2 + x2 = 1, so we have:sin2() + cos2()=1or s2 + c2 = 1or all angle .(cos(),sin())y=sin()(1,0)x=cos()
Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles .
Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities.
Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. The Trig-Hexagram
Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. The Trig-Hexagramthe CO-sidethe regular-side
Analytic TrigonometryThe Division Relations:
Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = II
Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),IIIIII
Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),IIIIII
Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII
Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A) The Reciprocal Relations
Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A) The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.
Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.Example B: sec(A) = 1/cos(A),
Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A)
Analytic Trigonometry Square-Sum Relations
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A) The identities from this hexagram are called theFundamental Identities.
Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A) The identities from this hexagram are called theFundamental Identities. Weassume these identities from here on and list the most important ones below.
Fundamental IdentitiesDivision Relationstan(A)=S/Ccot(A)=C/S
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/S
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations.
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations.sin(A) = cos(90 – A)sin(A) = cos(B)
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations.sin(A) = cos(90 – A)tan(A) = cot(90 – A)sin(A) = cos(B)tan(A) = cot(B)
Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations.sin(A) = cos(90 – A)tan(A) = cot(90 – A)sec(A) = csc(90 – A)sin(A) = cos(B)tan(A) = cot(B)sec(A) = csc(B)
Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11
Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).
Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).Therefore cosine is an even function and sine is an odd function.
Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).Therefore cosine is an even function and sine is an odd function. Since tangent and cotangent are quotients of sine and cosine, they are also odd functions.
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y) Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A))  Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A)  Frank Ma2006
Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A) = cos2(A) Frank Ma2006
Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine.
Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1)
Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1
Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation= tan2(x)
Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation= tan2(x) ; division relation = sin2(x)cos2(x)

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t3 analytic trigonometry and trig formulas

  • 2. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.
  • 3. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".
  • 4. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".
  • 5. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".
  • 6. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown.
  • 7. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown. (cos(),sin())y=sin()(1,0)x=cos()
  • 8. Analytic TrigonometryWe define the following reciprocals functions that are used often in science and engineering.csc() = 1/sin(), read as "cosecant of ".sec() = 1/cos(), read as "secant of ".cot() = 1/tan(), read as "cotangent of ".Let (x, y) be a point on the unit circle,thenx=cos() and y=sin()as shown. By Pythagorean Theorem,y2 + x2 = 1, so we have:sin2() + cos2()=1or s2 + c2 = 1or all angle .(cos(),sin())y=sin()(1,0)x=cos()
  • 9. Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles .
  • 10. Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities.
  • 11. Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. The Trig-Hexagram
  • 12. Analytic TrigonometryThe equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. The Trig-Hexagramthe CO-sidethe regular-side
  • 14. Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = II
  • 15. Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),IIIIII
  • 16. Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),IIIIII
  • 17. Analytic TrigonometryThe Division Relations:Start from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII
  • 18. Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A) The Reciprocal Relations
  • 19. Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A) The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.
  • 20. Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.Example B: sec(A) = 1/cos(A),
  • 21. Analytic TrigonometryThe Division RelationsStart from any function,going around the outside,we always haveI = II / III or I * III = IIExample A:tan(A) = sin(A)/cos(A),sec(A) = csc(A)/cot(A),sin(A)cot(A) = cos(A)IIIIII The Reciprocal RelationsStart from any function, going across diagonally, we always have I = II / III.Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A)
  • 22. Analytic Trigonometry Square-Sum Relations
  • 23. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.
  • 24. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1
  • 25. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)
  • 26. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)
  • 27. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A) The identities from this hexagram are called theFundamental Identities.
  • 28. Analytic Trigonometry Square-Sum RelationsFor each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.sin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A) The identities from this hexagram are called theFundamental Identities. Weassume these identities from here on and list the most important ones below.
  • 30. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/S
  • 31. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)
  • 32. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.
  • 33. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations.
  • 34. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations.sin(A) = cos(90 – A)sin(A) = cos(B)
  • 35. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations.sin(A) = cos(90 – A)tan(A) = cot(90 – A)sin(A) = cos(B)tan(A) = cot(B)
  • 36. Fundamental IdentitiesReciprocal Relationssec(A)=1/C csc(A)=1/Scot(A)=1/TDivision Relationstan(A)=S/Ccot(A)=C/SSquare-Sum Relationssin2(A) + cos2(A)=1tan2(A) + 1 = sec2(A)1 + cot2(A) = csc2(A)BAA and B are complementaryTwo angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations.sin(A) = cos(90 – A)tan(A) = cot(90 – A)sec(A) = csc(90 – A)sin(A) = cos(B)tan(A) = cot(B)sec(A) = csc(B)
  • 37. Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11
  • 38. Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).
  • 39. Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).Therefore cosine is an even function and sine is an odd function.
  • 40. Fundamental IdentitiesThe Negative Angle Relationscos(-A) = cos(A),sin(-A) = - sin(A)AA-A-A11Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).Therefore cosine is an even function and sine is an odd function. Since tangent and cotangent are quotients of sine and cosine, they are also odd functions.
  • 41. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.
  • 42. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) Frank Ma2006
  • 43. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) Frank Ma2006
  • 44. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 Frank Ma2006
  • 45. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 Frank Ma2006
  • 46. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Frank Ma2006
  • 47. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y) Frank Ma2006
  • 48. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A))  Frank Ma2006
  • 49. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A)  Frank Ma2006
  • 50. Notes on Square-Sum IdentitiesTerms in the square-sum identities may be rearranged and give other versions of the identities.Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A) = cos2(A) Frank Ma2006
  • 51. Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine.
  • 52. Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1)
  • 53. Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1
  • 54. Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation= tan2(x)
  • 55. Algebraic IdentitiesExample E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation= tan2(x) ; division relation = sin2(x)cos2(x)